Sure, let's solve the equation [tex]\((4x - 3)(x + 3) = -8\)[/tex] step by step.
1. Expand the left-hand side of the equation:
[tex]\[
(4x - 3)(x + 3) = 4x(x + 3) - 3(x + 3)
\][/tex]
Distribute [tex]\(4x\)[/tex] and [tex]\(-3\)[/tex]:
[tex]\[
4x^2 + 12x - 3x - 9 = 4x^2 + 9x - 9
\][/tex]
2. Rewrite the equation:
[tex]\[
4x^2 + 9x - 9 = -8
\][/tex]
3. Move all terms to one side to set the equation to zero:
[tex]\[
4x^2 + 9x - 9 + 8 = 0 \implies 4x^2 + 9x - 1 = 0
\][/tex]
4. Use the quadratic formula to solve for [tex]\(x\)[/tex]:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In this equation, [tex]\(a = 4\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = -1\)[/tex]. Plug these values into the formula:
[tex]\[
x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4}
\][/tex]
5. Simplify inside the square root:
[tex]\[
x = \frac{-9 \pm \sqrt{81 + 16}}{8}
\][/tex]
Which simplifies to:
[tex]\[
x = \frac{-9 \pm \sqrt{97}}{8}
\][/tex]
6. Separate the solutions:
[tex]\[
x = \frac{-9 + \sqrt{97}}{8} \quad \text{and} \quad x = \frac{-9 - \sqrt{97}}{8}
\][/tex]
So, the solutions to the equation [tex]\((4x - 3)(x + 3) = -8\)[/tex] are:
[tex]\[
x = \frac{-9 + \sqrt{97}}{8} \quad \text{and} \quad x = \frac{-9 - \sqrt{97}}{8}
\][/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the original equation.
