Solve for x if 2(5x + 2)2 = 48.

Question

x = StartFraction negative 2 + 2 StartRoot 24 EndRoot Over 5 EndFraction and x = StartFraction negative 2 minus 2 StartRoot 24 EndRoot Over 5 EndFraction
x = StartFraction negative 2 + 2 StartRoot 24 EndRoot Over 2 EndFraction and x = StartFraction negative 2 minus 2 StartRoot 24 EndRoot Over 2 EndFraction
x = StartFraction negative 2 + 2 StartRoot 6 EndRoot Over 5 EndFraction and x = StartFraction negative 2 minus 2 StartRoot 6 EndRoot Over 5 EndFraction
x = StartFraction negative 2 + 2 StartRoot 6 EndRoot Over 2 EndFraction and x = StartFraction negative 2 minus 2 StartRoot 6 EndRoot Over 2 EndFraction

Answer 1

To solve the equation $2(5x+2{)}^2=48$, let's start by isolating $(5x+2{)}^2$:

1. Divide both sides by 2: $$(5x+2{)}^2=\frac{48}{2}=24$$

2. Take the square root of both sides: $$5x+2=\pm \sqrt{24}$$ We can simplify $\sqrt{24}$: $$\sqrt{24}=\sqrt{4\times 6}=2\sqrt{6}$$ So we have: $$5x+2=2\sqrt{6}\text{and}5x+2=-2\sqrt{6}$$

3. Now, solve for $x$ in each case.

Case 1: $$5x+2=2\sqrt{6}$$ Subtract 2 from both sides: $$5x=2\sqrt{6}-2$$ Divide by 5: $$x=\frac{2\sqrt{6}-2}{5}$$

Case 2: $$5x+2=-2\sqrt{6}$$ Subtract 2 from both sides: $$5x=-2\sqrt{6}-2$$ Divide by 5: $$x=\frac{-2\sqrt{6}-2}{5}$$

Thus, the solutions for $x$ are: $$x=\frac{2\sqrt{6}-2}{5}\text{and}x=\frac{-2\sqrt{6}-2}{5}$$

These simplifications can also be written as: $$x=\frac{-2+2\sqrt{6}}{5}\text{and}x=\frac{-2-2\sqrt{6}}{5}$$

Therefore, the answers match with: $$x=\frac{-2+2\sqrt{6}}{5}\text{ and }x=\frac{-2-2\sqrt{6}}{5}$$

Hence, the correct answer corresponds to: $$x=\frac{-2+2\sqrt{6}}{5}\text{ and }x=\frac{-2-2\sqrt{6}}{5}$$