Solve x=12-sqrt{x^2-25}

$$x=12- \sqrt{ { x }^{ 2 } -25 }$$

Solve for x

$$x = \frac{169}{24} = 7\frac{1}{24} \approx 7.041666667$$

Solution Steps

Subtract 12 from both sides of the equation.

$$x-12=-\sqrt{x^{2}-25}$$

Square both sides of the equation.

$$\left(x-12\right)^{2}=\left(-\sqrt{x^{2}-25}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$x^{2}-24x+144=\left(-\sqrt{x^{2}-25}\right)^{2}$$

Expand \left(-\sqrt{x^{2}-25}\right)^{2}.

$$x^{2}-24x+144=\left(-1\right)^{2}\left(\sqrt{x^{2}-25}\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$x^{2}-24x+144=1\left(\sqrt{x^{2}-25}\right)^{2}$$

Calculate \sqrt{x^{2}-25} to the power of 2 and get x^{2}-25.

$$x^{2}-24x+144=1\left(x^{2}-25\right)$$

Use the distributive property to multiply 1 by x^{2}-25.

$$x^{2}-24x+144=x^{2}-25$$

Subtract x^{2} from both sides.

$$x^{2}-24x+144-x^{2}=-25$$

Combine x^{2} and -x^{2} to get 0.

$$-24x+144=-25$$

Subtract 144 from both sides.

$$-24x=-25-144$$

Subtract 144 from -25 to get -169.

$$-24x=-169$$

Divide both sides by -24.

$$x=\frac{-169}{-24}$$

Fraction \frac{-169}{-24} can be simplified to \frac{169}{24} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{169}{24}$$

Substitute \frac{169}{24} for x in the equation x=12-\sqrt{x^{2}-25}.

$$\frac{169}{24}=12-\sqrt{\left(\frac{169}{24}\right)^{2}-25}$$

Simplify. The value x=\frac{169}{24} satisfies the equation.

$$\frac{169}{24}=\frac{169}{24}$$

Equation x-12=-\sqrt{x^{2}-25} has a unique solution.

$$x=\frac{169}{24}$$