Question

Solve the following quadratic equation for all values of xx in simplest form.
4, left parenthesis, 3, x, minus, 9, right parenthesis, squared, minus, 9, equals, 7
4(3x−9)
2
−9=
7

Answer 1

To solve the quadratic equation given by

\[
4(3x - 9)^2 - 9 = 7,
\]

we'll first isolate the quadratic term.

1. Add 9 to both sides of the equation:

\[
4(3x - 9)^2 = 7 + 9
\]
\[
4(3x - 9)^2 = 16.
\]

2. Next, divide both sides by 4:

\[
(3x - 9)^2 = \frac{16}{4}
\]
\[
(3x - 9)^2 = 4.
\]

3. Now take the square root of both sides:

\[
3x - 9 = \pm 2.
\]

This gives us two cases to solve:

**Case 1:**

\[
3x - 9 = 2.
\]

Add 9 to both sides:

\[
3x = 2 + 9
\]
\[
3x = 11.
\]

Divide by 3:

\[
x = \frac{11}{3}.
\]

**Case 2:**

\[
3x - 9 = -2.
\]

Add 9 to both sides:

\[
3x = -2 + 9
\]
\[
3x = 7.
\]

Divide by 3:

\[
x = \frac{7}{3}.
\]

So the solutions to the quadratic equation are:

\[
x = \frac{11}{3} \quad \text{and} \quad x = \frac{7}{3}.
\]