Question
Solve for \( y \) : \[ (3 y+10)^{2}-64=0 \] INSTRUCTION: Where appropriate, give you answer as a simplified fraction.
Ask by Gross Schofield. in South Africa
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y = -6 \) or \( y = -\frac{2}{3} \)
Solution
Solve the equation \( (3y+10)^{2}-64=0 \).
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
\(\left(3y+10\right)^{2}-64=0\)
- step1: Add or subtract both sides:
\(\left(3y+10\right)^{2}=64\)
- step2: Simplify the expression:
\(3y+10=\pm \sqrt{64}\)
- step3: Simplify the expression:
\(3y+10=\pm 8\)
- step4: Separate into possible cases:
\(\begin{align}&3y+10=8\\&3y+10=-8\end{align}\)
- step5: Solve the equation:
\(\begin{align}&y=-\frac{2}{3}\\&y=-6\end{align}\)
- step6: Rewrite:
\(y_{1}=-6,y_{2}=-\frac{2}{3}\)
The solutions to the equation \((3y+10)^{2}-64=0\) are \(y=-6\) and \(y=-\frac{2}{3}\).
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Simplify this solution Beyond the Answer
To solve for \( y \) in the equation \( (3y + 10)^{2} - 64 = 0 \), we first add 64 to both sides: \[ (3y + 10)^{2} = 64 \] Next, we take the square root of both sides: \[ 3y + 10 = \pm 8 \] Now, we split this into two separate equations: 1. \( 3y + 10 = 8 \) 2. \( 3y + 10 = -8 \) For the first equation, we subtract 10 from both sides: \[ 3y = 8 - 10 \] \[ 3y = -2 \] Dividing by 3 gives: \[ y = -\frac{2}{3} \] For the second equation, we also subtract 10 from both sides: \[ 3y = -8 - 10 \] \[ 3y = -18 \] Dividing by 3 gives: \[ y = -6 \] Thus, the solutions are: \[ y = -\frac{2}{3} \quad \text{and} \quad y = -6 \]
