Solve z^2-12z+36=-64

$$z ^ { 2 } - 12 z + 36 = - 64$$

Solve for z

$$z=6+8i$$

$$z=6-8i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$z^{2}-12z+36=-64$$

Add 64 to both sides of the equation.

$$z^{2}-12z+36-\left(-64\right)=-64-\left(-64\right)$$

Subtracting -64 from itself leaves 0.

$$z^{2}-12z+36-\left(-64\right)=0$$

Subtract -64 from 36.

$$z^{2}-12z+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 100}}{2}$$

Square -12.

$$z=\frac{-\left(-12\right)±\sqrt{144-4\times 100}}{2}$$

Multiply -4 times 100.

$$z=\frac{-\left(-12\right)±\sqrt{144-400}}{2}$$

Add 144 to -400.

$$z=\frac{-\left(-12\right)±\sqrt{-256}}{2}$$

Take the square root of -256.

$$z=\frac{-\left(-12\right)±16i}{2}$$

The opposite of -12 is 12.

$$z=\frac{12±16i}{2}$$

Now solve the equation z=\frac{12±16i}{2} when ± is plus. Add 12 to 16i.

$$z=\frac{12+16i}{2}$$

Divide 12+16i by 2.

$$z=6+8i$$

Now solve the equation z=\frac{12±16i}{2} when ± is minus. Subtract 16i from 12.

$$z=\frac{12-16i}{2}$$

Divide 12-16i by 2.

$$z=6-8i$$

The equation is now solved.

$$z=6+8i z=6-8i$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$z^{2}-12z+36=-64$$

Factor z^{2}-12z+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-6\right)^{2}=-64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-6\right)^{2}}=\sqrt{-64}$$

Simplify.

$$z-6=8i z-6=-8i$$

Add 6 to both sides of the equation.

$$z=6+8i z=6-8i$$