Question

solving an equation involving a binomial squared
solve for x if $2(5x + 2)^2 = 48$.
\\(\boldsymbol{x = \frac{-2 + 2\sqrt{24}}{5}}\\) and \\(\boldsymbol{x = \frac{-2 - 2\sqrt{24}}{5}}\\)
\\(\boldsymbol{x = \frac{-2 + 2\sqrt{24}}{2}}\\) and \\(\boldsymbol{x = \frac{-2 - 2\sqrt{24}}{2}}\\)
\\(\boldsymbol{x = \frac{-2 + 2\sqrt{6}}{5}}\\) and \\(\boldsymbol{x = \frac{-2 - 2\sqrt{6}}{5}}\\)
\\(\boldsymbol{x = \frac{-2 + 2\sqrt{6}}{2}}\\) and \\(\boldsymbol{x = \frac{-2 - 2\sqrt{6}}{2}}\\)

Explanation:

Step1: Divide both sides by 2

To isolate the squared binomial, we divide each side of the equation \(2(5x + 2)^2 = 48\) by 2.
\[
\frac{2(5x + 2)^2}{2}=\frac{48}{2}
\]
This simplifies to \((5x + 2)^2 = 24\).

Step2: Take square roots

Take the square root of both sides. Remember that when we take the square root of a number, we consider both the positive and negative roots.
\[
5x + 2=\pm\sqrt{24}
\]

Step3: Isolate the x - term

Subtract 2 from both sides of the equation to isolate the term with \(x\).
\[
5x=-2\pm\sqrt{24}
\]

Step4: Simplify \(\sqrt{24}\)

We know that \(\sqrt{24}=\sqrt{4\times6} = 2\sqrt{6}\). So we can rewrite the equation as:
\[
5x=-2\pm2\sqrt{6}
\]

Step5: Solve for x

Divide both sides of the equation by 5 to solve for \(x\).
\[
x=\frac{-2\pm2\sqrt{6}}{5}
\]
This means \(x=\frac{-2 + 2\sqrt{6}}{5}\) and \(x=\frac{-2-2\sqrt{6}}{5}\)

Answer:

\(x=\frac{-2 + 2\sqrt{6}}{5}\) and \(x=\frac{-2-2\sqrt{6}}{5}\) (the third option: \(x=\frac{-2 + 2\sqrt{6}}{5}\) and \(x=\frac{-2-2\sqrt{6}}{5}\))