Question

Solve the following quadratic equation for all values of xx in simplest form.
5, left parenthesis, x, plus, 5, right parenthesis, squared, minus, 38, equals, minus, 18
5(x+5)
2
−38=
−18

Answer 1

To solve the quadratic equation given as:

\[ 5(x + 5)^2 - 38 = -18 \]

we'll start by simplifying the equation. First, we can move \(-18\) to the other side of the equation:

\[ 5(x + 5)^2 - 38 + 18 = 0 \]

This simplifies to:

\[ 5(x + 5)^2 - 20 = 0 \]

Next, we isolate the quadratic term by adding \(20\) to both sides:

\[ 5(x + 5)^2 = 20 \]

Now, divide both sides by \(5\):

\[ (x + 5)^2 = 4 \]

Next, we take the square root of both sides. Remember that this gives us two possible solutions:

\[ x + 5 = 2 \quad \text{or} \quad x + 5 = -2 \]

Now, we will solve for \(x\) in both cases.

1. \( x + 5 = 2 \)

Subtract \(5\) from both sides:

\[ x = 2 - 5 \]
\[ x = -3 \]

2. \( x + 5 = -2 \)

Subtract \(5\) from both sides:

\[ x = -2 - 5 \]
\[ x = -7 \]

Thus, the solutions to the equation are:

\[
x = -3 \quad \text{and} \quad x = -7
\]