Question

Solve the following quadratic equation for all values of xx in simplest form.
4, left parenthesis, x, squared, plus, 5, right parenthesis, minus, 17, equals, 12
4(x
2
+5)−17=
12

Answer 1

To solve the given quadratic equation, we start with:

\[ 4(x^2 + 5) - 17 = 12. \]

First, let's simplify the left side of the equation:

1. Distribute the 4:
\[
4x^2 + 20 - 17 = 12.
\]

2. Combine the constant terms (20 and -17):
\[
4x^2 + 3 = 12.
\]

Next, we need to isolate the \(x^2\) term:

3. Subtract 3 from both sides:
\[
4x^2 = 12 - 3,
\]
which simplifies to:
\[
4x^2 = 9.
\]

4. Now, divide both sides by 4:
\[
x^2 = \frac{9}{4}.
\]

5. To find \(x\), take the square root of both sides:
\[
x = \pm \sqrt{\frac{9}{4}}.
\]

The square root of a fraction can be simplified:
\[
x = \pm \frac{\sqrt{9}}{\sqrt{4}} = \pm \frac{3}{2}.
\]

Thus, the solutions for \(x\) are:
\[
x = \frac{3}{2} \quad \text{and} \quad x = -\frac{3}{2}.
\]

In conclusion, the values of \(x\) in simplest form are:
\[
\boxed{\frac{3}{2} \text{ and } -\frac{3}{2}}.
\]