Solve 5^2-x^2=6^2-(5-x^2)

$${ 5 }^{ 2 } - { x }^{ 2 } = { 6 }^{ 2 } -(5- { x }^{ 2 } )$$

Solve for x (complex solution)

$$x=-\sqrt{3}i\approx -0-1.732050808i$$

$$x=\sqrt{3}i\approx 1.732050808i$$

Steps by Finding Square Root

Calculate 5 to the power of 2 and get 25.

$$25-x^{2}=6^{2}-\left(5-x^{2}\right)$$

Calculate 6 to the power of 2 and get 36.

$$25-x^{2}=36-\left(5-x^{2}\right)$$

To find the opposite of 5-x^{2}, find the opposite of each term.

$$25-x^{2}=36-5+x^{2}$$

Subtract 5 from 36 to get 31.

$$25-x^{2}=31+x^{2}$$

Subtract x^{2} from both sides.

$$25-x^{2}-x^{2}=31$$

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

$$25-2x^{2}=31$$

Subtract 25 from both sides.

$$-2x^{2}=31-25$$

Subtract 25 from 31 to get 6.

$$-2x^{2}=6$$

Divide both sides by -2.

$$x^{2}=\frac{6}{-2}$$

Divide 6 by -2 to get -3.

$$x^{2}=-3$$

The equation is now solved.

$$x=\sqrt{3}i x=-\sqrt{3}i$$

Steps Using the Quadratic Formula

Calculate 5 to the power of 2 and get 25.

$$25-x^{2}=6^{2}-\left(5-x^{2}\right)$$

Calculate 6 to the power of 2 and get 36.

$$25-x^{2}=36-\left(5-x^{2}\right)$$

To find the opposite of 5-x^{2}, find the opposite of each term.

$$25-x^{2}=36-5+x^{2}$$

Subtract 5 from 36 to get 31.

$$25-x^{2}=31+x^{2}$$

Subtract 31 from both sides.

$$25-x^{2}-31=x^{2}$$

Subtract 31 from 25 to get -6.

$$-6-x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$-6-x^{2}-x^{2}=0$$

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

$$-6-2x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-2x^{2}-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 0 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-2\right)\left(-6\right)}}{2\left(-2\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2\right)\left(-6\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{0±\sqrt{8\left(-6\right)}}{2\left(-2\right)}$$

Multiply 8 times -6.

$$x=\frac{0±\sqrt{-48}}{2\left(-2\right)}$$

Take the square root of -48.

$$x=\frac{0±4\sqrt{3}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{0±4\sqrt{3}i}{-4}$$

Now solve the equation x=\frac{0±4\sqrt{3}i}{-4} when ± is plus.

$$x=-\sqrt{3}i$$

Now solve the equation x=\frac{0±4\sqrt{3}i}{-4} when ± is minus.

$$x=\sqrt{3}i$$

The equation is now solved.

$$x=-\sqrt{3}i x=\sqrt{3}i$$