Solve by completing the square. $\newline$ $y^2 - 2y = 31$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $y =$ _ or $y =$ _

Full solution

Q. Solve by completing the square.

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y^2 - 2y = 31

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Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

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y =

_____ or

y =

_____

Write Equation Form: Write the equation in the form of $y^2 + by = c$ . The given equation is $y^2 - 2y = 31$ .
Complete the Square: Add the square of half the coefficient of $y$ to both sides to complete the square. $\newline$ The coefficient of $y$ is $-2$ . Half of $-2$ is $-1$ , and $(-1)^2 = 1$ . Add $1$ to both sides. $\newline$ $y^2 - 2y + 1 = 31 + 1$ $\newline$ $y^2 - 2y + 1 = 32$
Factor Left Side: Factor the left side of the equation. $\newline$ The left side is a perfect square trinomial. $\newline$ $(y - 1)^2 = 32$
Take Square Root: Take the square root of both sides of the equation. $\newline$ $\sqrt{(y - 1)^2} = \pm\sqrt{32}$ $\newline$ $y - 1 = \pm\sqrt{32}$
Solve for y: Solve for y. $\newline$ Add $1$ to both sides of the equation to isolate $y$ . $\newline$ $y = 1 \pm \sqrt{32}$ $\newline$ Since $\sqrt{32} = \sqrt{(16 \cdot 2)} = 4\sqrt{2}$ , we can simplify further. $\newline$ $y = 1 \pm 4\sqrt{2}$
Approximate Square Root: Approximate the square root of $32$ to the nearest hundredth if necessary. $\newline$ $\sqrt{32} \approx 5.66$ (rounded to the nearest hundredth) $\newline$ So, $y \approx 1 \pm 5.66$ $\newline$ $y \approx 1 + 5.66$ or $y \approx 1 - 5.66$ $\newline$ $y \approx 6.66$ or $y \approx -4.66$