Campos Paul
06/07/2023 · Middle School
Find \( x \) such that the point \( (x, 0) \) is 10 units from \( (-1,-6) \). \( x=\square \) (Type an integer. Use a comma to separate answers as needed.)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Solve the equation \( \sqrt{(x+1)^2 + (-6)^2} = 10 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\sqrt{\left(x+1\right)^{2}+\left(-6\right)^{2}}=10\)
- step1: Add the terms:
\(\sqrt{x^{2}+2x+37}=10\)
- step2: Raise both sides to the \(2\)-th power\(:\)
\(\left(\sqrt{x^{2}+2x+37}\right)^{2}=10^{2}\)
- step3: Evaluate the power:
\(x^{2}+2x+37=100\)
- step4: Move the expression to the left side:
\(x^{2}+2x+37-100=0\)
- step5: Subtract the numbers:
\(x^{2}+2x-63=0\)
- step6: Factor the expression:
\(\left(x-7\right)\left(x+9\right)=0\)
- step7: Separate into possible cases:
\(\begin{align}&x-7=0\\&x+9=0\end{align}\)
- step8: Solve the equation:
\(\begin{align}&x=7\\&x=-9\end{align}\)
- step9: Check the solution:
\(\begin{align}&x=7\\&x=-9\end{align}\)
- step10: Rewrite:
\(x_{1}=-9,x_{2}=7\)
The solutions to the equation are \( x = -9 \) and \( x = 7 \).
Therefore, the values of \( x \) such that the point \( (x, 0) \) is 10 units from \( (-1,-6) \) are \( x = -9 \) and \( x = 7 \).
Quick Answer
\( x = -9 \) and \( x = 7 \).
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