Solve 5(5x-10)(2x-10)=20

$$5(5x-10)(2x-10)=20$$

Solve for x

$$x=\frac{\sqrt{265}}{10}+\frac{7}{2}\approx 5.12788206$$

$$x=-\frac{\sqrt{265}}{10}+\frac{7}{2}\approx 1.87211794$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5 by 5x-10.

$$\left(25x-50\right)\left(2x-10\right)=20$$

Use the distributive property to multiply 25x-50 by 2x-10 and combine like terms.

$$50x^{2}-350x+500=20$$

Subtract 20 from both sides.

$$50x^{2}-350x+500-20=0$$

Subtract 20 from 500 to get 480.

$$50x^{2}-350x+480=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 50 for a, -350 for b, and 480 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-350\right)±\sqrt{\left(-350\right)^{2}-4\times 50\times 480}}{2\times 50}$$

Square -350.

$$x=\frac{-\left(-350\right)±\sqrt{122500-4\times 50\times 480}}{2\times 50}$$

Multiply -4 times 50.

$$x=\frac{-\left(-350\right)±\sqrt{122500-200\times 480}}{2\times 50}$$

Multiply -200 times 480.

$$x=\frac{-\left(-350\right)±\sqrt{122500-96000}}{2\times 50}$$

Add 122500 to -96000.

$$x=\frac{-\left(-350\right)±\sqrt{26500}}{2\times 50}$$

Take the square root of 26500.

$$x=\frac{-\left(-350\right)±10\sqrt{265}}{2\times 50}$$

The opposite of -350 is 350.

$$x=\frac{350±10\sqrt{265}}{2\times 50}$$

Multiply 2 times 50.

$$x=\frac{350±10\sqrt{265}}{100}$$

Now solve the equation x=\frac{350±10\sqrt{265}}{100} when ± is plus. Add 350 to 10\sqrt{265}.

$$x=\frac{10\sqrt{265}+350}{100}$$

Divide 350+10\sqrt{265} by 100.

$$x=\frac{\sqrt{265}}{10}+\frac{7}{2}$$

Now solve the equation x=\frac{350±10\sqrt{265}}{100} when ± is minus. Subtract 10\sqrt{265} from 350.

$$x=\frac{350-10\sqrt{265}}{100}$$

Divide 350-10\sqrt{265} by 100.

$$x=-\frac{\sqrt{265}}{10}+\frac{7}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{265}}{10}+\frac{7}{2} x=-\frac{\sqrt{265}}{10}+\frac{7}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 5 by 5x-10.

$$\left(25x-50\right)\left(2x-10\right)=20$$

Use the distributive property to multiply 25x-50 by 2x-10 and combine like terms.

$$50x^{2}-350x+500=20$$

Subtract 500 from both sides.

$$50x^{2}-350x=20-500$$

Subtract 500 from 20 to get -480.

$$50x^{2}-350x=-480$$

Divide both sides by 50.

$$\frac{50x^{2}-350x}{50}=-\frac{480}{50}$$

Dividing by 50 undoes the multiplication by 50.

$$x^{2}+\left(-\frac{350}{50}\right)x=-\frac{480}{50}$$

Divide -350 by 50.

$$x^{2}-7x=-\frac{480}{50}$$

Reduce the fraction \frac{-480}{50} to lowest terms by extracting and canceling out 10.

$$x^{2}-7x=-\frac{48}{5}$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-\frac{48}{5}+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-\frac{48}{5}+\frac{49}{4}$$

Add -\frac{48}{5} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-7x+\frac{49}{4}=\frac{53}{20}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{53}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{53}{20}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{265}}{10} x-\frac{7}{2}=-\frac{\sqrt{265}}{10}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{\sqrt{265}}{10}+\frac{7}{2} x=-\frac{\sqrt{265}}{10}+\frac{7}{2}$$