Solve 80=-16x^2+64x

$$80=-16 { x }^{ 2 } +64x$$

Solve for x (complex solution)

$$x=2+i$$

$$x=2-i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-16x^{2}+64x=80$$

Subtract 80 from both sides.

$$-16x^{2}+64x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 64 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-64±\sqrt{64^{2}-4\left(-16\right)\left(-80\right)}}{2\left(-16\right)}$$

Square 64.

$$x=\frac{-64±\sqrt{4096-4\left(-16\right)\left(-80\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-64±\sqrt{4096+64\left(-80\right)}}{2\left(-16\right)}$$

Multiply 64 times -80.

$$x=\frac{-64±\sqrt{4096-5120}}{2\left(-16\right)}$$

Add 4096 to -5120.

$$x=\frac{-64±\sqrt{-1024}}{2\left(-16\right)}$$

Take the square root of -1024.

$$x=\frac{-64±32i}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-64±32i}{-32}$$

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

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

Divide -64+32i by -32.

$$x=2-i$$

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

$$x=\frac{-64-32i}{-32}$$

Divide -64-32i by -32.

$$x=2+i$$

The equation is now solved.

$$x=2-i x=2+i$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-16x^{2}+64x=80$$

Divide both sides by -16.

$$\frac{-16x^{2}+64x}{-16}=\frac{80}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{64}{-16}x=\frac{80}{-16}$$

Divide 64 by -16.

$$x^{2}-4x=\frac{80}{-16}$$

Divide 80 by -16.

$$x^{2}-4x=-5$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=-5+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-5+4$$

Add -5 to 4.

$$x^{2}-4x+4=-1$$

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}.

$$\left(x-2\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-1}$$

Simplify.

$$x-2=i x-2=-i$$

Add 2 to both sides of the equation.

$$x=2+i x=2-i$$