Exponential Equations in Quadratic Form: Equation, Type, Questions

Exponential Equations in Quadratic Form: Equation, Type, Questions

Edited By Komal Miglani | Updated on Jul 02, 2025 08:03 PM IST

Exponential equations and quadratic equations are important components of algebra, each with distinct characteristics and applications. However, there are scenarios where exponential equations can be transformed into a quadratic form, allowing the use of techniques from quadratic equations to solve them. Exponential equations can be expressed in quadratic form, providing insights into their properties, solution methods, and applications.

This Story also Contains
  1. Exponential Equations in Quadratic form
  2. Some special cases of the exponential equation
  3. Summary
  4. Solved Examples Based on Exponential Equations in Quadratic form:

Exponential Equations in Quadratic form

A polynomial equation in which the highest degree of a variable term is 2 is called a quadratic equation.

Standard form of quadratic equation is $a x^2+b x+c=0$

Where a, b, and c are constants (they may be real or imaginary) and called the coefficients of the equation and $a \neq 0$ (a is also called the leading coefficient).

Eg, $-5 x^2-3 x+2=0, x^2=0,(1+i) x^2-3 x+2 i=0$

As the degree of the quadratic polynomial is 2, so it always has 2 roots (number of real roots + number of imaginary roots = 2)

Roots of quadratic equation

The root of the quadratic equation is given by the formula:

$\begin{aligned} & x=\frac{-b \pm \sqrt{D}}{2 a} \\ & \text { or } \\ & x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}\end{aligned}$

Where D is called the discriminant of the quadratic equation, given by $D=b^2-4 a c$,

An equation of the form ax = b is known as an exponential equation, where

(i) $\mathrm{x} \in \phi$, if $\mathrm{b} \leq 0$
(ii) $x=\log _a b$, if $b>0, a \neq 0$
(iii) $\mathrm{x} \in \phi$, if $\mathrm{a}=1, \mathrm{~b} \neq 1$
(iv) $\mathrm{x} \in \mathrm{R}$, if $\mathrm{a}=1, \mathrm{~b}=1$ (since $1^{\mathrm{x}}=1 \Rightarrow 1=1, \mathrm{x} \in \mathrm{R}$ )

Some special cases of the exponential equation

1. Equation of the form $a^{f(x)}=1$, where $a>0$ and $a \neq 1$, then solve $f(x)=0$

For Example

The given equation is $7^{x^2+4 x+4}=1$
$\Rightarrow \mathrm{x}^2+4 \mathrm{x}+4=0 \quad\left[\because \mathrm{a}^0=1, \mathrm{a}\right.$ is constant $]$
$\Rightarrow(\mathrm{x}+2)(\mathrm{x}+2)=0 \Rightarrow \mathrm{x}=-2$

2. Equation of the form $f\left(a^x\right)=0$ then $f(t)=0$ where $t=a^x$

For example

The given equation is $4^x-3 \cdot 2^x-4=0$ equation is quadratic in $2^x$, so substitute $2^x=t$

$
\begin{aligned}
& \Rightarrow\left(2^{\mathrm{x}}\right)^2-3\left(2^{\mathrm{x}}\right)-4=0 \\
& \Rightarrow \mathrm{t}^2-3 \mathrm{t}-4=0 \\
& \Rightarrow(\mathrm{t}-4)(\mathrm{t}+1)=0 \\
& \Rightarrow \mathrm{t}=4, \mathrm{t}=-1
\end{aligned}
$

since, $\mathrm{t}=2^{\mathrm{x}}$

$
2^x=4 \Rightarrow 2^x=2^2 \Rightarrow x=2
$

and, $2^{\mathrm{x}}=-1$, No solution

Finally we get $x=2$

Summary

Exponential equations in quadratic form present an intriguing intersection of exponential and quadratic functions. By transforming exponential equations into quadratic form, we can apply familiar quadratic-solving techniques to find solutions. Understanding and mastering these transformations expand our ability to tackle a broader range of mathematical problems, enhancing both theoretical knowledge and practical problem-solving skills.

Recommended Video Based on Exponential Equations in Quadratic Form:

Solved Examples Based on Exponential Equations in Quadratic form:

Example 1: If the sum of all the roots of the equation $\mathrm{e}^{2 x}-11 \mathrm{e}^x-45 \mathrm{e}^{-x}+\frac{81}{2}=0$ is $\log _{\mathrm{e}} \mathrm{p}$, then $\mathrm{p}$ is equal to______________.

1) 45

2) 17

3) 46

4) 16

Solution

The equation can be rewritten as

$2 e^{3 x}-22 e^{2 x}+81 e^x-90=0$

Let the roots are $\alpha, \beta, \gamma$

Let $\mathrm{e}^{\mathrm{x}}=\mathrm{t}_{,} \mathrm{e}^\alpha=\mathrm{t}_1, \mathrm{e}^\beta=\mathrm{t}_2, \mathrm{e}^\gamma=\mathrm{t}_3$

$2 t^3-22 t^2+81 t-90=0$

Product of roots $=\mathrm{t}, \mathrm{t}_2 \mathrm{t}_3=\mathrm{e}^\alpha \cdot \mathrm{e}^\beta \cdot \mathrm{e}^\gamma=\frac{90}{2}=45$

$\Rightarrow \mathrm{e}^{\alpha+\beta+\gamma}=45$

$\Rightarrow \alpha+\beta+\gamma=\log _e 45$

So $\mathrm{p}=45$

Hence, the answer is 45.

Example 2: The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking, it was found that, in the observations, 20 was misread as 5 . Then, the correct variance is equal to_______.

1) 17

2) 43

3) 81

4) 12

Solution

$\frac{\displaystyle\sum_{i=1}^{14} x i+5}{15}=8$
$\Rightarrow \displaystyle\sum_{i=1}^{14} x i=115 \Rightarrow \displaystyle\sum_{i=1}^{11} x i+20=123$
$\Rightarrow$ Real Mean $=\frac{\displaystyle\sum_{i=1}^{14} x i+20}{15}=\frac{13}{15}$
$\frac{\displaystyle\sum_{i=1}^{14} x i^2+5^2}{15}-(8)^2=9 \Rightarrow \displaystyle\sum_{i=1}^{14} x i^2=1070$
$\begin{aligned} \text { So Real variance } & =\frac{\displaystyle\sum_{i=1}^{14} \times i^2+20^2}{15}-(9)^2 \\ & =\frac{1070+400}{15}-(9)^2 \\ & =\frac{1470}{15}-81\end{aligned}$
$=98-81=17$

Hence, the answer is 17.

Example 3: The number of real roots of the equation $e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^x+1=0$ is

1) 2

2) 4

3) 6

4) 1

Solution
$
\begin{aligned}
& e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^x+1=0 \\
& \left(\left(c^{3 x}\right)^2-2 e^{3 x}+1\right)-c^x\left(e^{3 x}-1\right)-12 e^{2 r}=0 \\
& \Rightarrow\left(e^{3 x}-1\right)^2-e^x\left(e^{3 x}-1\right)-12\left(e^x\right)^2=0 \\
& \text { Let } e^{3 r}-1=p \text { and } e^2=q \\
& =p^2-p q-12 q^2=0 \\
& \Rightarrow p^2-4 p q+3 p q-12 q^2=0 \\
& B(p-4 q)(p+3 q)=0 \\
& E p=4 q \text { or } p=-3 q \\
& \Rightarrow e^{3 x}-1=4 e^x \text { or } e^{3 x}-1=-3 e^x \\
& \operatorname{Let} e^x=t(\therefore t>0) \\
& \Rightarrow t^3-4 t-1=0 \text { or } t^3+3 t-1=0 \\
& \operatorname{Let} f(t)=t^3-4 t-1 \& \operatorname{Let} g(t)=t^3+3 t-1 \\
& f^{\prime}(t)=3 t^2-4 \& g^{\prime}(t)=3 t^2+3
\end{aligned}
$


$f(t)$ has one positive root $\mathrm{t}=1 \& g(t)$ has one positive root (say $\mathrm{t}_1$ )
So 2 solutions

Example 4: The number of real roots of the equation $e^{4 r}-e^{3 r}-4 e^{2 r}-e^r+1=0$ is equal to $\qquad$.

1) 2

2) 0

3) 1

4) 3

Solution

$
e^{4 x}-e^{2 x}-4 e^{2 x}-e^x+1=0
$

Let $e^x=1$

$
\begin{aligned}
& t^1-t^4-4 t^2-t+1=0 \\
& \Rightarrow t^2-t-4-\frac{1}{t}+\frac{1}{t^2}=0 \\
& \Rightarrow\left(t^2+\frac{1}{t^2}\right)-\left(t+\frac{1}{t}\right)-4=0 \\
& \Rightarrow\left(t+\frac{1}{t}\right)^2-\left(t+\frac{1}{t}\right)-6=0
\end{aligned}
$

Let $t+\frac{1}{t}=t$

$
\begin{aligned}
& \Rightarrow u^2-u-6=0 \\
& \Rightarrow(u-3)(u+2)=0 \\
& \Rightarrow u=3,-2 \\
& \Rightarrow t+\frac{1}{t}=3 \quad\left(\text { As } t+\frac{1}{t}=c^x+\frac{1}{c^x}>0\right) \\
& \Rightarrow t^2-3 t+1=0 \\
& \Rightarrow t=\frac{3 \pm \sqrt{9-4}}{2}=\frac{3+\sqrt{5}}{2}, \frac{3-\sqrt{5}}{2} \\
& \Rightarrow e^x=\frac{3+\sqrt{5}}{2}, e^t=\frac{3-\sqrt{5}}{2} \\
& \Rightarrow x=\ln \left(\frac{3+\sqrt{5}}{2}\right) \cdot \ln \left(\frac{3-\sqrt{5}}{2}\right)
\end{aligned}
$

Hence, the answer is the option 1.

Example 5: The number of real roots of the equation $\mathrm{e}^{4 x}+2 \mathrm{e}^{3 x}-\mathrm{e}^x-6=\mathrm{n}$ is:

1) 0

2) 1

3) 4

4) 2

Solution

$f(x)=e^{4 x}+2 e^{3 x}-e^x-6$
$f^{\prime}(x)=4 e^{4 x}+6 e^{3 x}-e^x=0$$\begin{aligned} & e^x\left(4 e^{3 x}+6 e^{2 x}-1\right)=0 \\ & e^x>0 ; g(x)=4 e^{3 x}+6 e^{2 x}-1 \\ & a(-\infty)=-1: g(\infty)=\infty\end{aligned}$
and $g(x)$ is always increasing function
$\Rightarrow g(x)$ will have exactly one real root
$\Rightarrow f^{\prime}(x)$ will have exactly one real root which will be point of minima
Also $f(0)=-4$$\begin{aligned} & f(-\infty)=-6 \\ & f(\infty)=\infty\end{aligned}$

$\Rightarrow f(x)=0$ will have exactly 1 real root.


Frequently Asked Questions (FAQs)

1. What are exponential equations in quadratic form?
Exponential equations in quadratic form are equations where the variable appears in the exponent, and the equation resembles a quadratic equation. These equations typically have the form a^(x^2) + b^x + c = 0, where a, b, and c are constants, and x is the variable.
2. How do exponential equations in quadratic form differ from regular quadratic equations?
While regular quadratic equations have the variable x as the base (ax^2 + bx + c = 0), exponential equations in quadratic form have x in the exponent. This fundamental difference requires different solving techniques and often involves logarithms.
3. What is the general approach to solving exponential equations in quadratic form?
The general approach involves substituting a new variable (usually y) for the exponential term (e.g., y = b^x), solving the resulting quadratic equation in y, and then using logarithms to solve for x.
4. Can all exponential equations be written in quadratic form?
No, not all exponential equations can be written in quadratic form. Only those that can be arranged to resemble the structure a^(x^2) + b^x + c = 0 or similar forms can be considered exponential equations in quadratic form.
5. How do you determine if an exponential equation is in quadratic form?
An exponential equation is in quadratic form if it can be written as a^(x^2) + b^x + c = 0, or if it can be manipulated into a similar form where the exponents of the variable terms differ by a factor of 2.
6. Why are logarithms often used in solving exponential equations in quadratic form?
Logarithms are used because they are the inverse operation of exponentiation. When we have an equation with variables in the exponent, taking the logarithm of both sides allows us to bring the variable down from the exponent, making it easier to solve.
7. How does the complexity of solving an exponential equation in quadratic form compare to solving a regular quadratic equation?
Solving exponential equations in quadratic form is generally more complex than solving regular quadratic equations. It often requires an additional step of using logarithms after solving the quadratic part, and may involve more algebraic manipulation.
8. Can the quadratic formula be directly applied to exponential equations in quadratic form?
The quadratic formula cannot be directly applied to the original exponential equation. However, after substituting a new variable for the exponential term, the quadratic formula can be used on the resulting quadratic equation.
9. What is the significance of the base in exponential equations in quadratic form?
The base in these equations is crucial as it determines the behavior of the function and affects the solution process. Common bases include e (natural exponential) and 2, but any positive number except 1 can be a base.
10. What role does the discriminant play in exponential equations in quadratic form?
The discriminant of the quadratic equation formed after substitution determines the nature and number of solutions to the original exponential equation. It helps in understanding whether the equation has real solutions and how many.
11. What is the importance of initial conditions in solving real-world problems involving exponential equations in quadratic form?
Initial conditions provide crucial information for determining the specific solution to a problem. They help in finding particular solutions from the general solution of the equation and are essential for accurately modeling real-world scenarios.
12. What are some real-world applications of exponential equations in quadratic form?
These equations can model various phenomena in physics, biology, and economics. For example, they may describe certain types of population growth, radioactive decay processes, or compound interest scenarios with changing rates.
13. How does changing the base of an exponential equation in quadratic form affect its solutions?
Changing the base doesn't change the number or existence of solutions, but it does affect the specific values of the solutions. The solutions will be scaled according to the new base, which is why logarithms with the appropriate base are used in solving.
14. Can exponential equations in quadratic form have complex solutions?
Yes, exponential equations in quadratic form can have complex solutions. This typically occurs when the quadratic equation formed after substitution has complex roots.
15. What is the relationship between the graph of an exponential function and its equation in quadratic form?
The graph of an exponential function in quadratic form often resembles a parabola, but with exponential growth or decay. The shape is determined by the coefficients and the base of the exponential terms.
16. How do you handle cases where the exponential equation in quadratic form has no real solutions?
If the equation has no real solutions, this will be evident after solving the quadratic equation formed by substitution. In such cases, we conclude that the original exponential equation has no real solutions either.
17. What is the importance of domain restrictions in exponential equations in quadratic form?
Domain restrictions are crucial because exponential functions with real bases are only defined for real inputs. Additionally, logarithms used in solving are only defined for positive arguments, which can further restrict the domain of solutions.
18. How do you verify solutions to exponential equations in quadratic form?
To verify solutions, substitute the found values of x back into the original equation. If the equation holds true (left side equals right side), the solution is correct.
19. Can exponential equations in quadratic form have infinitely many solutions?
While it's rare, certain special cases of exponential equations in quadratic form can have infinitely many solutions. This typically occurs when the equation can be reduced to a form where both sides are identical exponential expressions.
20. What is the significance of the y-intercept in exponential equations in quadratic form?
The y-intercept represents the value of the function when x = 0. In the context of exponential equations in quadratic form, it often corresponds to an initial value in real-world applications.
21. How does the concept of "change of base" apply to solving exponential equations in quadratic form?
The change of base formula for logarithms (log_a(x) = log_b(x) / log_b(a)) is often used when solving these equations, especially when dealing with bases other than e or 10, to convert to more manageable logarithms.
22. What are some common mistakes students make when solving exponential equations in quadratic form?
Common mistakes include forgetting to apply logarithms after solving the quadratic equation, incorrectly applying logarithm rules, neglecting domain restrictions, and failing to check solutions in the original equation.
23. How do you interpret negative exponents in the context of exponential equations in quadratic form?
Negative exponents in these equations represent reciprocals. For example, a^(-x) is equivalent to 1/(a^x). This interpretation is crucial for correctly manipulating and solving the equations.
24. Can exponential equations in quadratic form be solved graphically?
Yes, exponential equations in quadratic form can be solved graphically by plotting the exponential function and identifying where it intersects with the x-axis or another function, depending on how the equation is written.
25. What is the role of the natural exponential function (e^x) in solving exponential equations in quadratic form?
The natural exponential function often simplifies calculations due to its special properties, especially when using natural logarithms (ln). It's frequently used as a substitution to simplify more complex bases.
26. How do you approach exponential equations in quadratic form with fractional exponents?
Fractional exponents can be handled by rewriting them as roots. For example, x^(1/2) = √x. After this transformation, the equation can be solved using standard techniques for exponential equations in quadratic form.
27. What is the significance of the inflection point in the graph of an exponential function in quadratic form?
The inflection point, if it exists, represents where the graph changes from concave up to concave down or vice versa. It's important for understanding the behavior of the function and can be found by analyzing the second derivative.
28. How do exponential equations in quadratic form relate to logarithmic equations?
Exponential equations in quadratic form can often be converted to logarithmic equations by applying logarithms to both sides. This transformation is a key step in solving these equations and highlights the inverse relationship between exponential and logarithmic functions.
29. What role does the concept of "exponential growth" play in understanding exponential equations in quadratic form?
Exponential growth is central to these equations. The quadratic term in the exponent (x^2) can lead to extremely rapid growth or decay, which is important for modeling phenomena that change at an increasing or decreasing rate.
30. How do you determine the range of an exponential function in quadratic form?
The range is determined by analyzing the behavior of the function for all possible x values. For most exponential functions in quadratic form, the range is all positive real numbers, but it can be restricted based on the specific equation and any transformations applied.
31. What is the importance of understanding asymptotic behavior in exponential equations in quadratic form?
Asymptotic behavior helps in understanding the long-term trends of the function. For these equations, it's crucial to recognize how the function behaves as x approaches positive or negative infinity, which can provide insights into the nature of the solutions and the function's overall behavior.
32. How do you handle exponential equations in quadratic form with multiple bases?
For equations with multiple bases, a common approach is to convert all exponential terms to a single base using logarithm properties. This often involves using the change of base formula to express all terms in terms of a common logarithm (usually natural log or log base 10).
33. What is the significance of the discriminant in the quadratic equation formed after substitution in solving exponential equations in quadratic form?
The discriminant of the quadratic equation formed after substitution determines the nature of the solutions to the original exponential equation. A positive discriminant indicates two distinct real solutions, zero discriminant indicates one repeated real solution, and a negative discriminant indicates complex solutions.
34. How do you approach exponential equations in quadratic form where the variable appears both in the base and the exponent?
These equations are particularly challenging and often require advanced techniques. One approach is to use logarithms to bring down the exponent, then use substitution to create a new equation that can be solved using standard methods.
35. What is the relationship between exponential equations in quadratic form and exponential regression in data analysis?
Exponential equations in quadratic form can be used in exponential regression to model data that shows exponential growth or decay with a changing rate. The quadratic term in the exponent allows for more flexibility in fitting curves to data that doesn't follow simple exponential patterns.
36. How do you determine if an exponential equation in quadratic form represents growth or decay?
The nature of growth or decay is determined by the signs and magnitudes of the coefficients in the exponent. Generally, if the coefficient of x^2 is positive, the function will ultimately show growth as x increases, while a negative coefficient will lead to decay.
37. How do you handle exponential equations in quadratic form with absolute value terms?
Equations with absolute value terms often require considering multiple cases. The equation is typically split into two separate equations - one for when the expression inside the absolute value is non-negative, and another for when it's negative. Each case is then solved separately.
38. What role does the concept of "doubling time" or "half-life" play in exponential equations in quadratic form?
While doubling time and half-life are more straightforward in simple exponential equations, in quadratic form they become variable. The quadratic term in the exponent means that the rate of doubling or halving changes over time, which can model more complex growth or decay scenarios.
39. How do you approach exponential equations in quadratic form that involve trigonometric functions?
These equations often require a combination of techniques from exponential, trigonometric, and quadratic equation solving. One approach is to use substitution to simplify the equation, then apply appropriate trigonometric identities before solving the resulting equation.
40. What is the significance of the point of intersection between two exponential functions in quadratic form?
The point of intersection represents where two different exponential processes have the same value. In real-world applications, this could indicate when two different growth or decay processes reach parity, which can be crucial in decision-making scenarios.
41. How do you handle exponential equations in quadratic form with complex coefficients?
Equations with complex coefficients are solved using similar techniques as those with real coefficients, but require careful handling of complex numbers throughout the process. The solutions may be complex, and interpretation of these solutions depends on the context of the problem.
42. What is the importance of understanding the concept of "e" as the base of natural logarithms in solving exponential equations in quadratic form?
The number e is particularly useful in these equations because of its special properties in calculus and its relationship with natural logarithms. Using e as a base often simplifies calculations and is especially useful when the equation involves rates of change or continuous growth/decay.
43. How do you approach exponential equations in quadratic form that involve parametric expressions?
Parametric expressions add an extra layer of complexity. The approach typically involves solving the equation in terms of the parameters, which may result in conditions on the parameters for solutions to exist. The final solution often expresses x in terms of these parameters.
44. What is the significance of the maximum or minimum point in an exponential function in quadratic form?
The maximum or minimum point, if it exists, represents a turning point in the function's behavior. It's crucial for understanding the overall shape of the function and can represent critical values in real-world applications, such as peak population in a growth model.
45. How do you handle exponential equations in quadratic form that involve logarithmic terms?
These equations often require careful application of logarithm properties. One approach is to use the properties of logarithms to simplify the equation, potentially converting logarithmic terms to exponential ones, before applying standard solving techniques.
46. What is the role of dimensional analysis in solving real-world problems involving exponential equations in quadratic form?
Dimensional analysis ensures that the units in the equation are consistent throughout the problem-solving process. It's particularly important in exponential equations because exponents must be dimensionless, which can provide clues about the correct form of the equation.
47. How do you approach exponential equations in quadratic form that involve infinite series?
These complex equations often require advanced techniques from calculus, such as Taylor series expansions. The approach typically involves truncating the series to a finite number of terms, solving the resulting approximate equation, and then considering the error introduced by the truncation.
48. What is the significance of the rate of change in exponential functions in quadratic form?
The rate of change in these functions is itself an exponential function, which leads to accelerating growth or decay. Understanding this is crucial for accurately interpreting the behavior of the function, especially in modeling real-world phenomena with changing rates.
49. How do you handle exponential equations in quadratic form with piecewise-defined functions?
Piecewise-defined functions require solving the equation separately for each piece of the function's domain. The overall solution is then a combination of these separate solutions, taking care to consider the domain restrictions for each piece.
50. What is the importance of understanding the relationship between exponential and polynomial functions when dealing with exponential equations in quadratic form?
Understanding this relationship is crucial because exponential equations in quadratic form combine aspects of both exponential and polynomial (specifically quadratic) functions. This hybrid nature affects the function's behavior, solution methods, and interpretations, often leading to more complex and interesting phenomena than either exponential or quadratic functions alone.

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