Q. \(13 – x^{2} = -10\).

Q: What is the first step to solve 13 - x^2 = -10?.

Subtract 13 from both sides to get -x^2 = -23. Then multiply by -1 to obtain x^2 = 23.

Q: How do I find x from x^2 = 23?

Take square roots: x = \pm\sqrt{23}. Both the positive and negative roots are solutions because squaring either gives 23.

Q: Are the solutions real or complex?

Since 23>0, the square roots \pm\sqrt{23} are real numbers. No complex solutions arise here.

Q: Could I have made an algebraic mistake like losing a sign?

Common error is forgetting to multiply by -1 after -x^2=-23. That step yields x^2=23; missing the sign flips the result incorrectly.

Q: Should I check the solutions in the original equation?

Yes. Substitute x=\pm\sqrt{23} into 13-x^2: 13-(\pm\sqrt{23})^2=13-23=-10. Both satisfy the original equation.

Q: What are the decimal approximations of the solutions?.

\sqrt{23}\approx 4.795831523 , so x\approx 4.7958 or x\approx -4.7958 .

Q: How does this look graphically?

Graph y=13-x^2 (a downward parabola) and y=-10 (a horizontal line). Their intersections occur at x=\pm\sqrt{23}, the two solutions.

Answer

Start with the equation.

\(13-x^2=-10\)

Add \(x^2\) to both sides.

\(13=-10+x^2\)

Add \(10\) to both sides.

\(23=x^2\)

So:

\(x^2=23\)

Take the square root of both sides.

\(x=\pm\sqrt{23}\)

Final result: \(x=\pm\sqrt{23}\)

Detailed Explanation

Problem

Solve the equation

\[13 – x^{2} = -10\]

Step 1 — Isolate the squared term

Subtract 13 from both sides to move the constant term to the right-hand side. This gives

\[13 – x^{2} – 13 = -10 – 13\]

Simplify each side:

\[-x^{2} = -23\]

Step 2 — Remove the negative sign

Multiply both sides by −1 to make the coefficient of \(x^{2}\) positive:

\[(-1)\cdot(-x^{2}) = (-1)\cdot(-23)\]

\[x^{2} = 23\]

Step 3 — Take square roots

Take the square root of both sides. Remember that taking the square root of both sides yields two solutions, positive and negative, because both a number and its negative square to the same positive value:

\[x = \pm\sqrt{23}\]

Step 4 — Check the solutions

Substitute \(x = \sqrt{23}\):

\[13 – (\sqrt{23})^{2} = 13 – 23 = -10\]

Substitute \(x = -\sqrt{23}\):

\[13 – (-\sqrt{23})^{2} = 13 – 23 = -10\]

Both values satisfy the original equation.

Final answer

\[x = \pm\sqrt{23}\]

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Algebra FAQs

What is the first step to solve \(13 - x^2 = -10\)?.

Subtract 13 from both sides to get \(-x^2 = -23\). Then multiply by \(-1\) to obtain \(x^2 = 23\).

How do I find \(x\) from \(x^2 = 23\)?

Take square roots: \(x = \pm\sqrt{23}\). Both the positive and negative roots are solutions because squaring either gives 23.

Are the solutions real or complex?

Since \(23>0\), the square roots \(\pm\sqrt{23}\) are real numbers. No complex solutions arise here.

Could I have made an algebraic mistake like losing a sign?

Common error is forgetting to multiply by \(-1\) after \(-x^2=-23\). That step yields \(x^2=23\); missing the sign flips the result incorrectly.

Should I check the solutions in the original equation?

Yes. Substitute \(x=\pm\sqrt{23}\) into \(13-x^2\): \(13-(\pm\sqrt{23})^2=13-23=-10\). Both satisfy the original equation.

Can I solve this by factoring?

What are the decimal approximations of the solutions?.

\( \sqrt{23}\approx 4.795831523 \), so \( x\approx 4.7958 \) or \( x\approx -4.7958 \).

How does this look graphically?

Graph \(y=13-x^2\) (a downward parabola) and \(y=-10\) (a horizontal line). Their intersections occur at \(x=\pm\sqrt{23}\), the two solutions.

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