Solve (x-5)(x-25)=1700

$$(x-5)(x-25)=1700$$

Solve for x

$$x=30\sqrt{2}+15\approx 57.426406871$$

$$x=15-30\sqrt{2}\approx -27.426406871$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-5 by x-25 and combine like terms.

$$x^{2}-30x+125=1700$$

Subtract 1700 from both sides.

$$x^{2}-30x+125-1700=0$$

Subtract 1700 from 125 to get -1575.

$$x^{2}-30x-1575=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -30 for b, and -1575 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-1575\right)}}{2}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\left(-1575\right)}}{2}$$

Multiply -4 times -1575.

$$x=\frac{-\left(-30\right)±\sqrt{900+6300}}{2}$$

Add 900 to 6300.

$$x=\frac{-\left(-30\right)±\sqrt{7200}}{2}$$

Take the square root of 7200.

$$x=\frac{-\left(-30\right)±60\sqrt{2}}{2}$$

The opposite of -30 is 30.

$$x=\frac{30±60\sqrt{2}}{2}$$

Now solve the equation x=\frac{30±60\sqrt{2}}{2} when ± is plus. Add 30 to 60\sqrt{2}.

$$x=\frac{60\sqrt{2}+30}{2}$$

Divide 60\sqrt{2}+30 by 2.

$$x=30\sqrt{2}+15$$

Now solve the equation x=\frac{30±60\sqrt{2}}{2} when ± is minus. Subtract 60\sqrt{2} from 30.

$$x=\frac{30-60\sqrt{2}}{2}$$

Divide 30-60\sqrt{2} by 2.

$$x=15-30\sqrt{2}$$

The equation is now solved.

$$x=30\sqrt{2}+15 x=15-30\sqrt{2}$$

Steps for Completing the Square

Use the distributive property to multiply x-5 by x-25 and combine like terms.

$$x^{2}-30x+125=1700$$

Subtract 125 from both sides.

$$x^{2}-30x=1700-125$$

Subtract 125 from 1700 to get 1575.

$$x^{2}-30x=1575$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=1575+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=1575+225$$

Add 1575 to 225.

$$x^{2}-30x+225=1800$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=1800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{1800}$$

Simplify.

$$x-15=30\sqrt{2} x-15=-30\sqrt{2}$$

Add 15 to both sides of the equation.

$$x=30\sqrt{2}+15 x=15-30\sqrt{2}$$