Question

question 7 of 10
which of the following are solutions to the equation below?
check all that apply.
$(3x - 5)^2 = 19$

a. $x = \sqrt{19} + \frac{5}{3}$
b. $x = \frac{-\sqrt{19} + 5}{3}$
c. $x = -\frac{\sqrt{14}}{3}$
d. $x = \frac{\sqrt{14}}{3}$
e. $x = \frac{\sqrt{19} + 5}{3}$
f. $x = -\sqrt{19} + \frac{5}{3}$

Explanation:

Step1: Take square root of both sides

To solve \((3x - 5)^2 = 19\), we first take the square root of both sides. Remember that if \(y^2 = a\), then \(y=\pm\sqrt{a}\). So, applying this to our equation, we get:
\(3x - 5=\pm\sqrt{19}\)

Step2: Solve for \(x\) in each case

Case 1: When \(3x - 5=\sqrt{19}\)

We add 5 to both sides:
\(3x=\sqrt{19}+ 5\)
Then divide both sides by 3:
\(x=\frac{\sqrt{19}+5}{3}\) (This is option E)

Case 2: When \(3x - 5=-\sqrt{19}\)

We add 5 to both sides:
\(3x = -\sqrt{19}+5\)
Then divide both sides by 3:
\(x=\frac{-\sqrt{19}+5}{3}\) (This is option B)

Answer:

B. \( x = \frac{-\sqrt{19}+5}{3} \), E. \( x = \frac{\sqrt{19}+5}{3} \)