Certainly! Let's solve the quadratic equation step-by-step.
Given quadratic equation:
[tex]\[ 4x^2 - 9x - 9 = 0 \][/tex]
This is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = 4 \)[/tex], [tex]\( b = -9 \)[/tex], and [tex]\( c = -9 \)[/tex].
To solve this equation, we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, let's identify the components that we need:
1. Calculate [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
b^2 - 4ac = (-9)^2 - 4(4)(-9)
\][/tex]
[tex]\[
= 81 + 144
\][/tex]
[tex]\[
= 225
\][/tex]
2. Calculate the square root of [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
\sqrt{225} = 15
\][/tex]
3. Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\(\sqrt{225}\)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-(-9) \pm 15}{2 \times 4}
\][/tex]
[tex]\[
= \frac{9 \pm 15}{8}
\][/tex]
Now, we have two possible solutions:
1. When [tex]\( + \)[/tex] is used:
[tex]\[
x = \frac{9 + 15}{8} = \frac{24}{8} = 3
\][/tex]
2. When [tex]\( - \)[/tex] is used:
[tex]\[
x = \frac{9 - 15}{8} = \frac{-6}{8} = -\frac{3}{4}
\][/tex]
Hence, the solutions to the quadratic equation [tex]\( 4x^2 - 9x - 9 = 0 \)[/tex] are:
[tex]\[
x = -\frac{3}{4} \quad \text{or} \quad x = 3
\][/tex]
