Question

Solve the following quadratic equation for all values of xx in simplest form.
4, left parenthesis, x, squared, minus, 19, right parenthesis, minus, 15, equals, minus, 10
4(x
2
−19)−15=
−10

Answer 1

To solve the equation \( 4(x^2 - 19) - 15 = -10 \), we can start by simplifying it step by step.

First, let's isolate the term inside the parentheses by adding 15 to both sides:

\[
4(x^2 - 19) - 15 + 15 = -10 + 15
\]
\[
4(x^2 - 19) = 5
\]

Next, divide both sides by 4 to get rid of the coefficient in front of the parentheses:

\[
x^2 - 19 = \frac{5}{4}
\]

Now, we will add 19 to both sides:

\[
x^2 = 19 + \frac{5}{4}
\]

To combine these terms, we need a common denominator. The number 19 can be expressed as \( \frac{76}{4} \):

\[
x^2 = \frac{76}{4} + \frac{5}{4}
\]
\[
x^2 = \frac{81}{4}
\]

Now, take the square root of both sides:

\[
x = \pm \sqrt{\frac{81}{4}}
\]
\[
x = \pm \frac{\sqrt{81}}{\sqrt{4}}
\]
\[
x = \pm \frac{9}{2}
\]

Thus, the solutions to the quadratic equation \( 4(x^2 - 19) - 15 = -10 \) are:

\[
x = \frac{9}{2} \quad \text{and} \quad x = -\frac{9}{2}
\]