Solve 5x^2-165x+250=0

$$5 { x }^{ 2 } -165x+250=0$$

Solve for x

$$x = \frac{\sqrt{889} + 33}{2} \approx 31.408051516$$

$$x = \frac{33 - \sqrt{889}}{2} \approx 1.591948484$$

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.

$$5x^{2}-165x+250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -165 for b, and 250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-165\right)±\sqrt{\left(-165\right)^{2}-4\times 5\times 250}}{2\times 5}$$

Square -165.

$$x=\frac{-\left(-165\right)±\sqrt{27225-4\times 5\times 250}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-165\right)±\sqrt{27225-20\times 250}}{2\times 5}$$

Multiply -20 times 250.

$$x=\frac{-\left(-165\right)±\sqrt{27225-5000}}{2\times 5}$$

Add 27225 to -5000.

$$x=\frac{-\left(-165\right)±\sqrt{22225}}{2\times 5}$$

Take the square root of 22225.

$$x=\frac{-\left(-165\right)±5\sqrt{889}}{2\times 5}$$

The opposite of -165 is 165.

$$x=\frac{165±5\sqrt{889}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{165±5\sqrt{889}}{10}$$

Now solve the equation x=\frac{165±5\sqrt{889}}{10} when ± is plus. Add 165 to 5\sqrt{889}.

$$x=\frac{5\sqrt{889}+165}{10}$$

Divide 165+5\sqrt{889} by 10.

$$x=\frac{\sqrt{889}+33}{2}$$

Now solve the equation x=\frac{165±5\sqrt{889}}{10} when ± is minus. Subtract 5\sqrt{889} from 165.

$$x=\frac{165-5\sqrt{889}}{10}$$

Divide 165-5\sqrt{889} by 10.

$$x=\frac{33-\sqrt{889}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{889}+33}{2} x=\frac{33-\sqrt{889}}{2}$$

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.

$$5x^{2}-165x+250=0$$

Subtract 250 from both sides of the equation.

$$5x^{2}-165x+250-250=-250$$

Subtracting 250 from itself leaves 0.

$$5x^{2}-165x=-250$$

Divide both sides by 5.

$$\frac{5x^{2}-165x}{5}=-\frac{250}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{165}{5}\right)x=-\frac{250}{5}$$

Divide -165 by 5.

$$x^{2}-33x=-\frac{250}{5}$$

Divide -250 by 5.

$$x^{2}-33x=-50$$

Divide -33, the coefficient of the x term, by 2 to get -\frac{33}{2}. Then add the square of -\frac{33}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-33x+\left(-\frac{33}{2}\right)^{2}=-50+\left(-\frac{33}{2}\right)^{2}$$

Square -\frac{33}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-33x+\frac{1089}{4}=-50+\frac{1089}{4}$$

Add -50 to \frac{1089}{4}.

$$x^{2}-33x+\frac{1089}{4}=\frac{889}{4}$$

Factor x^{2}-33x+\frac{1089}{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}.

$$\left(x-\frac{33}{2}\right)^{2}=\frac{889}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{33}{2}\right)^{2}}=\sqrt{\frac{889}{4}}$$

Simplify.

$$x-\frac{33}{2}=\frac{\sqrt{889}}{2} x-\frac{33}{2}=-\frac{\sqrt{889}}{2}$$

Add \frac{33}{2} to both sides of the equation.

$$x=\frac{\sqrt{889}+33}{2} x=\frac{33-\sqrt{889}}{2}$$