Question

Determine the values of \(x\) that satisfy the equation \(\left(x^{2}+3 x-9\right)^{2 x-8}=1\)

Short answer

\(x = -5, -4, 2\)

Step-by-step solution

Identify the Properties of Equation

To find the values of \(x\) that satisfy \(\left(x^{2}+3 x-9\right)^{2 x-8}=1\), recognize that an expression raised to any power can equal 1 if the base is 1 or -1 and the exponent is even, or if the exponent is 0.

Case 1: Base Equals 1

For the expression to equal 1, the base \(\left(x^{2}+3 x-9\right)\) can be 1. Solve for \(x\) by setting up the equation: \[ x^{2} + 3x - 9 = 1 \] \Simplify this to get: \[ x^{2} + 3x - 10 = 0 \] \Factorize: \[ (x+5)(x-2) = 0 \] \Therefore, \(x = -5\) or \(x = 2\).

Case 2: Base Equals -1 and Exponent is Even

For the expression to equal 1, the base \(\left(x^{2}+3 x-9\right)\) can also be -1 provided the exponent \(2x-8\) is even. Solve for \(x\) by setting up the equation: \[ x^{2} + 3x - 9 = -1 \] \Simplify this to get: \[ x^{2} + 3x - 8 = 0 \] \Factorize: \[ (x+4)(x-2) = 0 \] \Therefore, \(x = -4\) or \(x = 2\).

Check if Exponent is Even

For the base to be -1 and to satisfy the equation, the exponent \(2x-8\) must be even. Substitute \(x = -4\): \[ 2(-4)-8 = -16 \] (even) \Substitute \(x = 2\): \[ 2(2)-8 = -4 \] (even) \Both values of \(x\) yield an even exponent.

Solution Set

Combine the solutions from both cases: \(x = -5, -4, 2\)

Key concepts

Exponential Equations

When dealing with exponential equations like \(\big(x^{2}+3x-9\big)^{2x-8}=1\), it is important to identify the properties of such equations. Here, the problem requires us to understand under which conditions the equation holds true. Exponential equations can have solutions based on the base and the exponent. The core concept here is recognizing when an exponent is zero or when a base equals 1 or -1, under specific conditions.

Remember that any number raised to the power of zero equals 1. Therefore, the equation can hold if the exponent is zero. Similarly, a number raised to an even exponent can be 1 if the base is -1, as \((-1)^2 = 1\).

Here, we solve the equation by setting up the conditions for the base and exponent to fit these properties. The understanding of exponential behavior helps immensely in identifying correct steps. Make sure to always start by identifying these properties whenever you encounter exponential equations.

Factoring Quadratic Equations

Factoring quadratic equations is a crucial step in solving equations like the one in this example. When faced with an equation in the form \( x^{2} + 3x - 10 = 0 \), we need to factorize it to find the values of \(x\).

Factoring involves breaking down the quadratic equation into a product of simpler binomials. In the example given, \( x^{2} + 3x - 10 = 0 \) is factorized as \( (x+5)(x-2) = 0 \). This means that the solutions can be found by setting each binomial equal to zero: \( x+5 = 0 \) and \( x-2 = 0 \), giving us \( x = -5 \) and \( x = 2 \) respectively.

The same approach is applied to another part of the problem where the equation becomes \( x^{2} + 3x - 8 = 0 \). This is factorized into \( (x+4)(x-2) = 0 \), resulting in solutions \( x = -4 \) and \( x = 2 \).

Understanding how to factor quadratic equations is vital as it simplifies finding the roots significantly. Look for values that add up to the coefficient of the middle term (in this case 3) and multiply to the constant term (in this case -10 and -8). Practice makes perfect in recognizing these patterns.

Checking Solutions

After solving equations, it's critical to check that the proposed solutions satisfy the original equation. This ensures there were no mistakes in the computation.

For this particular problem, after identifying potential solutions \( x = -5, -4, 2 \), you need to verify them against the original equation \( \big(x^{2}+3x-9\big)^{2x-8}=1 \). Start by substituting each value of \(x\) into the original equation and check if the equation holds true.

For \( x = -5 \), we substitute and check: \begin{aligned} \( -5^{2} + 3(-5) - 9 \)= 1, and \(2(-5)-8 \) is even. \text{All conditions are satisfied.}
\text{For } \( x = -4 \), There’s the same process to check conditions with the base and exponent. If they are met, this solution is valid.
Finally, for \( x = 2 \), since \( (2)^{2}+3(2)-9 \)= 1, all conditions are met.

Checking solutions in this sequential manner ensures that your solutions are accurate and fit all required conditions of the equation. Always revalidate your solutions to confirm their correctness.