Question
Solve the quadratic equation by completing the square. \[ x^{2}+2 x=63 \] \( x=\square \) (Simplify your answer. Type an exact answer, using radicals and \( i \) as needed. Use a comma to separate answers as needed.)
Ask by Mullins Curry. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -9, 7 \)
Solution
Solve the equation \( x^{2}+2x=63 \).
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
\(x^{2}+2x=63\)
- step1: Move the expression to the left side:
\(x^{2}+2x-63=0\)
- step2: Factor the expression:
\(\left(x-7\right)\left(x+9\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&x-7=0\\&x+9=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=7\\&x=-9\end{align}\)
- step5: Rewrite:
\(x_{1}=-9,x_{2}=7\)
The solutions to the quadratic equation \(x^{2}+2x=63\) by completing the square are \(x=-9\) and \(x=7\).
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Simplify this solution Extra Insights
To solve the quadratic equation \( x^{2}+2x=63 \) by completing the square, we first rearrange the equation: \[ x^2 + 2x - 63 = 0. \] Next, we'll isolate the quadratic and linear terms: \[ x^2 + 2x = 63. \] Now, to complete the square, take half of the coefficient of \( x \) (which is 2), square it (resulting in 1), and add it to both sides: \[ x^2 + 2x + 1 = 63 + 1, \] \[ (x + 1)^2 = 64. \] Take the square root of both sides: \[ x + 1 = \pm 8. \] Now, solve for \( x \): 1. \( x + 1 = 8 \) leads to \( x = 7 \), 2. \( x + 1 = -8 \) leads to \( x = -9 \). Thus, the solutions to the equation are: \( x = 7, -9 \).
