Solve -16x^2+64x-60=60

$$-16 { x }^{ 2 } +64x-60=60$$

Solve for x (complex solution)

$$x=\frac{\sqrt{14}i}{2}+2\approx 2+1.870828693i$$

$$x=-\frac{\sqrt{14}i}{2}+2\approx 2-1.870828693i$$

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.

$$-16x^{2}+64x-60=60$$

Subtract 60 from both sides of the equation.

$$-16x^{2}+64x-60-60=60-60$$

Subtracting 60 from itself leaves 0.

$$-16x^{2}+64x-60-60=0$$

Subtract 60 from -60.

$$-16x^{2}+64x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 64 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-64±\sqrt{64^{2}-4\left(-16\right)\left(-120\right)}}{2\left(-16\right)}$$

Square 64.

$$x=\frac{-64±\sqrt{4096-4\left(-16\right)\left(-120\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-64±\sqrt{4096+64\left(-120\right)}}{2\left(-16\right)}$$

Multiply 64 times -120.

$$x=\frac{-64±\sqrt{4096-7680}}{2\left(-16\right)}$$

Add 4096 to -7680.

$$x=\frac{-64±\sqrt{-3584}}{2\left(-16\right)}$$

Take the square root of -3584.

$$x=\frac{-64±16\sqrt{14}i}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-64±16\sqrt{14}i}{-32}$$

Now solve the equation x=\frac{-64±16\sqrt{14}i}{-32} when ± is plus. Add -64 to 16i\sqrt{14}.

$$x=\frac{-64+16\sqrt{14}i}{-32}$$

Divide -64+16i\sqrt{14} by -32.

$$x=-\frac{\sqrt{14}i}{2}+2$$

Now solve the equation x=\frac{-64±16\sqrt{14}i}{-32} when ± is minus. Subtract 16i\sqrt{14} from -64.

$$x=\frac{-16\sqrt{14}i-64}{-32}$$

Divide -64-16i\sqrt{14} by -32.

$$x=\frac{\sqrt{14}i}{2}+2$$

The equation is now solved.

$$x=-\frac{\sqrt{14}i}{2}+2 x=\frac{\sqrt{14}i}{2}+2$$

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.

$$-16x^{2}+64x-60=60$$

Add 60 to both sides of the equation.

$$-16x^{2}+64x-60-\left(-60\right)=60-\left(-60\right)$$

Subtracting -60 from itself leaves 0.

$$-16x^{2}+64x=60-\left(-60\right)$$

Subtract -60 from 60.

$$-16x^{2}+64x=120$$

Divide both sides by -16.

$$\frac{-16x^{2}+64x}{-16}=\frac{120}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{64}{-16}x=\frac{120}{-16}$$

Divide 64 by -16.

$$x^{2}-4x=\frac{120}{-16}$$

Reduce the fraction \frac{120}{-16} to lowest terms by extracting and canceling out 8.

$$x^{2}-4x=-\frac{15}{2}$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{15}{2}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-\frac{15}{2}+4$$

Add -\frac{15}{2} to 4.

$$x^{2}-4x+4=-\frac{7}{2}$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=-\frac{7}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-\frac{7}{2}}$$

Simplify.

$$x-2=\frac{\sqrt{14}i}{2} x-2=-\frac{\sqrt{14}i}{2}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{14}i}{2}+2 x=-\frac{\sqrt{14}i}{2}+2$$