Solve -x^2+90x-75=20

$$- x ^ { 2 } + 90 x - 75 = 20$$

Solve for x

$$x=\sqrt{1930}+45\approx 88.931765273$$

$$x=45-\sqrt{1930}\approx 1.068234727$$

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.

$$-x^{2}+90x-75=20$$

Subtract 20 from both sides of the equation.

$$-x^{2}+90x-75-20=20-20$$

Subtracting 20 from itself leaves 0.

$$-x^{2}+90x-75-20=0$$

Subtract 20 from -75.

$$-x^{2}+90x-95=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 90 for b, and -95 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\left(-1\right)\left(-95\right)}}{2\left(-1\right)}$$

Square 90.

$$x=\frac{-90±\sqrt{8100-4\left(-1\right)\left(-95\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-90±\sqrt{8100+4\left(-95\right)}}{2\left(-1\right)}$$

Multiply 4 times -95.

$$x=\frac{-90±\sqrt{8100-380}}{2\left(-1\right)}$$

Add 8100 to -380.

$$x=\frac{-90±\sqrt{7720}}{2\left(-1\right)}$$

Take the square root of 7720.

$$x=\frac{-90±2\sqrt{1930}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-90±2\sqrt{1930}}{-2}$$

Now solve the equation x=\frac{-90±2\sqrt{1930}}{-2} when ± is plus. Add -90 to 2\sqrt{1930}.

$$x=\frac{2\sqrt{1930}-90}{-2}$$

Divide -90+2\sqrt{1930} by -2.

$$x=45-\sqrt{1930}$$

Now solve the equation x=\frac{-90±2\sqrt{1930}}{-2} when ± is minus. Subtract 2\sqrt{1930} from -90.

$$x=\frac{-2\sqrt{1930}-90}{-2}$$

Divide -90-2\sqrt{1930} by -2.

$$x=\sqrt{1930}+45$$

The equation is now solved.

$$x=45-\sqrt{1930} x=\sqrt{1930}+45$$

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.

$$-x^{2}+90x-75=20$$

Add 75 to both sides of the equation.

$$-x^{2}+90x-75-\left(-75\right)=20-\left(-75\right)$$

Subtracting -75 from itself leaves 0.

$$-x^{2}+90x=20-\left(-75\right)$$

Subtract -75 from 20.

$$-x^{2}+90x=95$$

Divide both sides by -1.

$$\frac{-x^{2}+90x}{-1}=\frac{95}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{90}{-1}x=\frac{95}{-1}$$

Divide 90 by -1.

$$x^{2}-90x=\frac{95}{-1}$$

Divide 95 by -1.

$$x^{2}-90x=-95$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-95+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-95+2025$$

Add -95 to 2025.

$$x^{2}-90x+2025=1930$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=1930$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{1930}$$

Simplify.

$$x-45=\sqrt{1930} x-45=-\sqrt{1930}$$

Add 45 to both sides of the equation.

$$x=\sqrt{1930}+45 x=45-\sqrt{1930}$$