Solve the following equation for y. {:[y+6=sqrt(2y^(2)+72)],[y=]:}

Question

Solve the following equation for  y.  {:[y+6=sqrt(2y^(2)+72)],[y=]:}

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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: (y + 6)^2 = (sqrt(2y^2 + 72))^2Expand left side: Now we perform the squaring on both sides: (y + 6)^2 = 2y^2 + 72Simplify the equation: We expand the left side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2: y^2 + 2*6*y + 6^2 = 2y^2 + 72Bring all terms to one side: Simplify the equation: y^2 + 12y + 36 = 2y^2 + 72Rearrange 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: y^2 + 12y + 36 - y^2 - 72 = 2y^2 + 72 - y^2 - 72Factor the quadratic equation: Simplify the equation by combining like terms: 12y - 36 = y^2Set factors equal to zero: Rearrange the equation to set it to a standard quadratic form: y^2 - 12y + 36 = 0Check potential solution: Now we factor the quadratic equation: (y - 6)(y - 6) = 0Set each factor equal to zero and solve for y: y - 6 = 0 y = 6We 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: 6 + 6 = sqrt(2*6^2 + 72) 12 = sqrt(72 + 72) 12 = sqrt(144) 12 = 12 The solution checks out.

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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Solve the following equation for y y y.\newliney+6=2y2+72y=□ \begin{array}{l} y+6=\sqrt{2 y^{2}+72} \\ y=\square \end{array} y+6=2y2+72​y=□​

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