Question

Solve the following quadratic equation for all values of xx in simplest form.
5, left parenthesis, 5, x, minus, 9, right parenthesis, squared, minus, 39, equals, 41
5(5x−9)
2
−39=
41

Answer 1

We start with the quadratic equation:

\[
5(5x - 9)^2 - 39 = 41
\]

First, we can isolate the squared term. Add 39 to both sides:

\[
5(5x - 9)^2 = 80
\]

Next, divide both sides by 5:

\[
(5x - 9)^2 = 16
\]

Now, we take the square root of both sides. Remember to consider both the positive and negative roots:

\[
5x - 9 = 4 \quad \text{or} \quad 5x - 9 = -4
\]

Let's solve each equation separately.

**For the first equation:**

\[
5x - 9 = 4
\]
Add 9 to both sides:

\[
5x = 13
\]

Now, divide by 5:

\[
x = \frac{13}{5}
\]

**For the second equation:**

\[
5x - 9 = -4
\]
Add 9 to both sides:

\[
5x = 5
\]

Now, divide by 5:

\[
x = 1
\]

In summary, the solutions to the equation \(5(5x - 9)^2 - 39 = 41\) are:

\[
\boxed{\frac{13}{5} \text{ and } 1}
\]