To solve the quadratic equation [tex]\(4(x^2 - 6) - 3 = 9\)[/tex] for all values of [tex]\(x\)[/tex], follow these steps:
1. Simplify the equation:
Start by adding 3 to both sides of the equation to isolate the term involving the quadratic expression.
[tex]\[
4(x^2 - 6) - 3 + 3 = 9 + 3
\][/tex]
This simplifies to:
[tex]\[
4(x^2 - 6) = 12
\][/tex]
2. Isolate the quadratic term:
Divide both sides of the equation by 4 to further simplify.
[tex]\[
\frac{4(x^2 - 6)}{4} = \frac{12}{4}
\][/tex]
This simplifies to:
[tex]\[
x^2 - 6 = 3
\][/tex]
3. Solve for [tex]\(x^2\)[/tex]:
Add 6 to both sides of the equation to isolate [tex]\(x^2\)[/tex].
[tex]\[
x^2 - 6 + 6 = 3 + 6
\][/tex]
This simplifies to:
[tex]\[
x^2 = 9
\][/tex]
4. Take the square root of both sides:
To solve for [tex]\(x\)[/tex], take the square root of both sides of the equation. Remember that the square root of a number can be both positive and negative.
[tex]\[
x = \pm \sqrt{9}
\][/tex]
[tex]\[
x = \pm 3
\][/tex]
Therefore, the solutions to the equation are:
[tex]\[
x = 3 \quad \text{and} \quad x = -3
\][/tex]
