Solve (-2x+100)(x-30)=150

$$( - 2 x + 100 ) ( x - 30 ) = 150$$

Solve for x

$$x=35$$

$$x=45$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -2x+100 by x-30 and combine like terms.

$$-2x^{2}+160x-3000=150$$

Subtract 150 from both sides.

$$-2x^{2}+160x-3000-150=0$$

Subtract 150 from -3000 to get -3150.

$$-2x^{2}+160x-3150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 160 for b, and -3150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-160±\sqrt{160^{2}-4\left(-2\right)\left(-3150\right)}}{2\left(-2\right)}$$

Square 160.

$$x=\frac{-160±\sqrt{25600-4\left(-2\right)\left(-3150\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-160±\sqrt{25600+8\left(-3150\right)}}{2\left(-2\right)}$$

Multiply 8 times -3150.

$$x=\frac{-160±\sqrt{25600-25200}}{2\left(-2\right)}$$

Add 25600 to -25200.

$$x=\frac{-160±\sqrt{400}}{2\left(-2\right)}$$

Take the square root of 400.

$$x=\frac{-160±20}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-160±20}{-4}$$

Now solve the equation x=\frac{-160±20}{-4} when ± is plus. Add -160 to 20.

$$x=-\frac{140}{-4}$$

Divide -140 by -4.

$$x=35$$

Now solve the equation x=\frac{-160±20}{-4} when ± is minus. Subtract 20 from -160.

$$x=-\frac{180}{-4}$$

Divide -180 by -4.

$$x=45$$

The equation is now solved.

$$x=35 x=45$$

Steps for Completing the Square

Use the distributive property to multiply -2x+100 by x-30 and combine like terms.

$$-2x^{2}+160x-3000=150$$

Add 3000 to both sides.

$$-2x^{2}+160x=150+3000$$

Add 150 and 3000 to get 3150.

$$-2x^{2}+160x=3150$$

Divide both sides by -2.

$$\frac{-2x^{2}+160x}{-2}=\frac{3150}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{160}{-2}x=\frac{3150}{-2}$$

Divide 160 by -2.

$$x^{2}-80x=\frac{3150}{-2}$$

Divide 3150 by -2.

$$x^{2}-80x=-1575$$

Divide -80, the coefficient of the x term, by 2 to get -40. Then add the square of -40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-80x+\left(-40\right)^{2}=-1575+\left(-40\right)^{2}$$

Square -40.

$$x^{2}-80x+1600=-1575+1600$$

Add -1575 to 1600.

$$x^{2}-80x+1600=25$$

Factor x^{2}-80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-40\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-40=5 x-40=-5$$

Add 40 to both sides of the equation.

$$x=45 x=35$$