Solve by completing the square. $\newline$ $p^2 - 8p = 23$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $p =$ _ or $p =$ _

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Algebra 2

Solve a quadratic equation by completing the square

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Q. Solve by completing the square.

\newline

p^2 - 8p = 23

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

\newline

p =

_____ or

p =

_____

Rewrite Equation: Rewrite the equation in the form of $p^2 + bp = c$ . We have the equation $p^2 - 8p = 23$ . To complete the square, we need to move the constant term to the right side of the equation. Add $23$ to both sides to isolate the $p^2$ and $p$ terms. $p^2 - 8p + 0 = 23 + 0$ $p^2 - 8p = 23$
Add Constant Term: Choose the number to add to both sides to complete the square. $\newline$ To complete the square, we need to add $(b/2)^2$ to both sides, where $b$ is the coefficient of $p$ . In this case, $b = -8$ . $\newline$ $(-8/2)^2 = (-4)^2 = 16$ $\newline$ Add $16$ to both sides of the equation. $\newline$ $p^2 − 8p + 16 = 23 + 16$ $\newline$ $p^2 − 8p + 16 = 39$
Choose Number to Add: Factor the left side of the equation. $\newline$ The left side of the equation is now a perfect square trinomial. $\newline$ $(p - 4)^2 = 39$
Factor Left Side: Take the square root of both sides of the equation. $\newline$ To solve for $p$ , take the square root of both sides. $\newline$ $\sqrt{(p - 4)^2} = \pm\sqrt{39}$ $\newline$ $p - 4 = \pm\sqrt{39}$
Take Square Root: Solve for $p$ . $\newline$ To isolate $p$ , add $4$ to both sides of the equation. $\newline$ $p - 4 + 4 = \pm\sqrt{39} + 4$ $\newline$ p = $4 \pm \sqrt{39}$
Solve for $p$ : Approximate the square root of $39$ to the nearest hundredth. $\sqrt{39}$ is approximately $6.24$ (rounded to the nearest hundredth). So, $p \approx 4 \pm 6.24$
Approximate Square Root: Find the two values of $p$ . $p \approx 4 + 6.24$ implies $p \approx 10.24$ . $p \approx 4 - 6.24$ implies $p \approx -2.24$ .Values of $p$ : $10.24$ , $-2.24$