Solve the following equation for $y$ . $\newline$ $\begin{array}{l} y+6=\sqrt{2 y^{2}+72} \\ y=\square \end{array}$

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Grade 8

Relationship between squares and square roots

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Q. Solve the following equation for

y

\newline

\begin{array}{l} y+6=\sqrt{2 y^{2}+72} \\ y=\square \end{array}

Isolate square root: First, we need to isolate the square root on one side of the equation. We already have that, so we can move to the next step.
Square both sides: Next, we square both sides of the equation to eliminate the square root. This gives us: $\newline$ (y + \(6)^ $2$ = (\sqrt{ $2$ y^ $2$ + $72$ })^ $2$
Expand left side: Now we perform the squaring on both sides: $\newline$ $(y + 6)^2 = 2y^2 + 72$
Simplify the equation: We expand the left side of the equation using the formula $(a + b)^2 = a^2 + 2ab + b^2$ : $\newline$ $y^2 + 2 \cdot 6 \cdot y + 6^2 = 2y^2 + 72$
Bring all terms to one side: Simplify the equation: $y^2 + 12y + 36 = 2y^2 + 72$
Rearrange to standard form: We want to bring all terms to one side to set the equation to zero. Subtract $y^2$ and $72$ from both sides: $\newline$ $y^2 + 12y + 36 - y^2 - 72 = 2y^2 + 72 - y^2 - 72$
Factor the quadratic equation: Simplify the equation by combining like terms: $12y - 36 = y^2$
Set factors equal to zero: Rearrange the equation to set it to a standard quadratic form: $\newline$ $y^2 - 12y + 36 = 0$
Check potential solution: Now we factor the quadratic equation: $\newline$ $(y - 6)(y - 6) = 0$
Check potential solution: Now we factor the quadratic equation: $\newline$ $(y - 6)(y - 6) = 0$ Set each factor equal to zero and solve for $y$ : $\newline$ $y - 6 = 0$ $\newline$ $y = 6$
Check potential solution: Now we factor the quadratic equation: $\newline$ $(y - 6)(y - 6) = 0$ Set each factor equal to zero and solve for y: $\newline$ $y - 6 = 0$ $\newline$ $y = 6$ We have found a potential solution for $y$ . However, we must check it in the original equation to ensure it does not create any mathematical inconsistencies, such as a negative number under the square root.
Check potential solution: Now we factor the quadratic equation: $\newline$ $(y - 6)(y - 6) = 0$ Set each factor equal to zero and solve for $y$ : $\newline$ $y - 6 = 0$ $\newline$ $y = 6$ We have found a potential solution for $y$ . However, we must check it in the original equation to ensure it does not create any mathematical inconsistencies, such as a negative number under the square root. Check the solution $y = 6$ in the original equation: $\newline$ $6 + 6 = \sqrt{2 \cdot 6^2 + 72}$ $\newline$ $12 = \sqrt{72 + 72}$ $\newline$ $12 = \sqrt{144}$ $\newline$ $12 = 12$ $\newline$ The solution checks out.