Solve 16k^2-32k+16=18k^2+360

$$16 k ^ { 2 } - 32 k + 16 = 18 k ^ { 2 } + 360$$

Solve for k

$$k=-8+6\sqrt{3}i\approx -8+10.392304845i$$

$$k=-6\sqrt{3}i-8\approx -8-10.392304845i$$

Steps Using the Quadratic Formula

Subtract 18k^{2} from both sides.

$$16k^{2}-32k+16-18k^{2}=360$$

Combine 16k^{2} and -18k^{2} to get -2k^{2}.

$$-2k^{2}-32k+16=360$$

Subtract 360 from both sides.

$$-2k^{2}-32k+16-360=0$$

Subtract 360 from 16 to get -344.

$$-2k^{2}-32k-344=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -32 for b, and -344 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-2\right)\left(-344\right)}}{2\left(-2\right)}$$

Square -32.

$$k=\frac{-\left(-32\right)±\sqrt{1024-4\left(-2\right)\left(-344\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$k=\frac{-\left(-32\right)±\sqrt{1024+8\left(-344\right)}}{2\left(-2\right)}$$

Multiply 8 times -344.

$$k=\frac{-\left(-32\right)±\sqrt{1024-2752}}{2\left(-2\right)}$$

Add 1024 to -2752.

$$k=\frac{-\left(-32\right)±\sqrt{-1728}}{2\left(-2\right)}$$

Take the square root of -1728.

$$k=\frac{-\left(-32\right)±24\sqrt{3}i}{2\left(-2\right)}$$

The opposite of -32 is 32.

$$k=\frac{32±24\sqrt{3}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$k=\frac{32±24\sqrt{3}i}{-4}$$

Now solve the equation k=\frac{32±24\sqrt{3}i}{-4} when ± is plus. Add 32 to 24i\sqrt{3}.

$$k=\frac{32+24\sqrt{3}i}{-4}$$

Divide 32+24i\sqrt{3} by -4.

$$k=-6\sqrt{3}i-8$$

Now solve the equation k=\frac{32±24\sqrt{3}i}{-4} when ± is minus. Subtract 24i\sqrt{3} from 32.

$$k=\frac{-24\sqrt{3}i+32}{-4}$$

Divide 32-24i\sqrt{3} by -4.

$$k=-8+6\sqrt{3}i$$

The equation is now solved.

$$k=-6\sqrt{3}i-8 k=-8+6\sqrt{3}i$$

Steps for Completing the Square

Subtract 18k^{2} from both sides.

$$16k^{2}-32k+16-18k^{2}=360$$

Combine 16k^{2} and -18k^{2} to get -2k^{2}.

$$-2k^{2}-32k+16=360$$

Subtract 16 from both sides.

$$-2k^{2}-32k=360-16$$

Subtract 16 from 360 to get 344.

$$-2k^{2}-32k=344$$

Divide both sides by -2.

$$\frac{-2k^{2}-32k}{-2}=\frac{344}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$k^{2}+\left(-\frac{32}{-2}\right)k=\frac{344}{-2}$$

Divide -32 by -2.

$$k^{2}+16k=\frac{344}{-2}$$

Divide 344 by -2.

$$k^{2}+16k=-172$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}+16k+8^{2}=-172+8^{2}$$

Square 8.

$$k^{2}+16k+64=-172+64$$

Add -172 to 64.

$$k^{2}+16k+64=-108$$

Factor k^{2}+16k+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k+8\right)^{2}=-108$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+8\right)^{2}}=\sqrt{-108}$$

Simplify.

$$k+8=6\sqrt{3}i k+8=-6\sqrt{3}i$$

Subtract 8 from both sides of the equation.

$$k=-8+6\sqrt{3}i k=-6\sqrt{3}i-8$$