Solve left(2sqrt{5}-sqrt{5}xright)left(5-xright)frac{2sqrt{5}}{5}=4

$$\left( 2 \sqrt{ 5 } - \sqrt{ 5 } x \right) \left( 5-x \right) \frac{ 2 \sqrt{ 5 } }{ 5 } = 4$$

Solve for x

$$x = \frac{7 - \sqrt{17}}{2} \approx 1.438447187$$

$$x = \frac{\sqrt{17} + 7}{2} \approx 5.561552813$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$\left(2\sqrt{5}-\sqrt{5}x\right)\left(5-x\right)\times 2\sqrt{5}=20$$

Use the distributive property to multiply 2\sqrt{5}-\sqrt{5}x by 5-x.

$$\left(5\left(2\sqrt{5}-\sqrt{5}x\right)-\left(2\sqrt{5}-\sqrt{5}x\right)x\right)\times 2\sqrt{5}=20$$

Use the distributive property to multiply 5\left(2\sqrt{5}-\sqrt{5}x\right)-\left(2\sqrt{5}-\sqrt{5}x\right)x by 2.

$$\left(10\left(2\sqrt{5}-\sqrt{5}x\right)-2\left(2\sqrt{5}-\sqrt{5}x\right)x\right)\sqrt{5}=20$$

Use the distributive property to multiply 10\left(2\sqrt{5}-\sqrt{5}x\right)-2\left(2\sqrt{5}-\sqrt{5}x\right)x by \sqrt{5}.

$$10\left(2\sqrt{5}-\sqrt{5}x\right)\sqrt{5}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

Subtract 20 from both sides.

$$10\left(2\sqrt{5}-\sqrt{5}x\right)\sqrt{5}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}-20=0$$

Use the distributive property to multiply 10 by 2\sqrt{5}-\sqrt{5}x.

$$\left(20\sqrt{5}-10\sqrt{5}x\right)\sqrt{5}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}-20=0$$

Use the distributive property to multiply 20\sqrt{5}-10\sqrt{5}x by \sqrt{5}.

$$20\left(\sqrt{5}\right)^{2}-10x\left(\sqrt{5}\right)^{2}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}-20=0$$

The square of \sqrt{5} is 5.

$$20\times 5-10x\left(\sqrt{5}\right)^{2}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}-20=0$$

Multiply 20 and 5 to get 100.

$$100-10x\left(\sqrt{5}\right)^{2}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}-20=0$$

The square of \sqrt{5} is 5.

$$100-10x\times 5-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}-20=0$$

Multiply -10 and 5 to get -50.

$$100-50x-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}-20=0$$

Use the distributive property to multiply -2 by 2\sqrt{5}-\sqrt{5}x.

$$100-50x+\left(-4\sqrt{5}+2\sqrt{5}x\right)x\sqrt{5}-20=0$$

Use the distributive property to multiply -4\sqrt{5}+2\sqrt{5}x by x.

$$100-50x+\left(-4\sqrt{5}x+2\sqrt{5}x^{2}\right)\sqrt{5}-20=0$$

Use the distributive property to multiply -4\sqrt{5}x+2\sqrt{5}x^{2} by \sqrt{5}.

$$100-50x-4x\left(\sqrt{5}\right)^{2}+2x^{2}\left(\sqrt{5}\right)^{2}-20=0$$

The square of \sqrt{5} is 5.

$$100-50x-4x\times 5+2x^{2}\left(\sqrt{5}\right)^{2}-20=0$$

Multiply -4 and 5 to get -20.

$$100-50x-20x+2x^{2}\left(\sqrt{5}\right)^{2}-20=0$$

The square of \sqrt{5} is 5.

$$100-50x-20x+2x^{2}\times 5-20=0$$

Multiply 2 and 5 to get 10.

$$100-50x-20x+10x^{2}-20=0$$

Combine -50x and -20x to get -70x.

$$100-70x+10x^{2}-20=0$$

Subtract 20 from 100 to get 80.

$$80-70x+10x^{2}=0$$

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.

$$10x^{2}-70x+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -70 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 10\times 80}}{2\times 10}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 10\times 80}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-70\right)±\sqrt{4900-40\times 80}}{2\times 10}$$

Multiply -40 times 80.

$$x=\frac{-\left(-70\right)±\sqrt{4900-3200}}{2\times 10}$$

Add 4900 to -3200.

$$x=\frac{-\left(-70\right)±\sqrt{1700}}{2\times 10}$$

Take the square root of 1700.

$$x=\frac{-\left(-70\right)±10\sqrt{17}}{2\times 10}$$

The opposite of -70 is 70.

$$x=\frac{70±10\sqrt{17}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{70±10\sqrt{17}}{20}$$

Now solve the equation x=\frac{70±10\sqrt{17}}{20} when ± is plus. Add 70 to 10\sqrt{17}.

$$x=\frac{10\sqrt{17}+70}{20}$$

Divide 70+10\sqrt{17} by 20.

$$x=\frac{\sqrt{17}+7}{2}$$

Now solve the equation x=\frac{70±10\sqrt{17}}{20} when ± is minus. Subtract 10\sqrt{17} from 70.

$$x=\frac{70-10\sqrt{17}}{20}$$

Divide 70-10\sqrt{17} by 20.

$$x=\frac{7-\sqrt{17}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+7}{2} x=\frac{7-\sqrt{17}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$\left(2\sqrt{5}-\sqrt{5}x\right)\left(5-x\right)\times 2\sqrt{5}=20$$

Use the distributive property to multiply 2\sqrt{5}-\sqrt{5}x by 5-x.

$$\left(5\left(2\sqrt{5}-\sqrt{5}x\right)-\left(2\sqrt{5}-\sqrt{5}x\right)x\right)\times 2\sqrt{5}=20$$

Use the distributive property to multiply 5\left(2\sqrt{5}-\sqrt{5}x\right)-\left(2\sqrt{5}-\sqrt{5}x\right)x by 2.

$$\left(10\left(2\sqrt{5}-\sqrt{5}x\right)-2\left(2\sqrt{5}-\sqrt{5}x\right)x\right)\sqrt{5}=20$$

Use the distributive property to multiply 10\left(2\sqrt{5}-\sqrt{5}x\right)-2\left(2\sqrt{5}-\sqrt{5}x\right)x by \sqrt{5}.

$$10\left(2\sqrt{5}-\sqrt{5}x\right)\sqrt{5}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

Use the distributive property to multiply 10 by 2\sqrt{5}-\sqrt{5}x.

$$\left(20\sqrt{5}-10\sqrt{5}x\right)\sqrt{5}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

Use the distributive property to multiply 20\sqrt{5}-10\sqrt{5}x by \sqrt{5}.

$$20\left(\sqrt{5}\right)^{2}-10x\left(\sqrt{5}\right)^{2}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

The square of \sqrt{5} is 5.

$$20\times 5-10x\left(\sqrt{5}\right)^{2}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

Multiply 20 and 5 to get 100.

$$100-10x\left(\sqrt{5}\right)^{2}-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

The square of \sqrt{5} is 5.

$$100-10x\times 5-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

Multiply -10 and 5 to get -50.

$$100-50x-2\left(2\sqrt{5}-\sqrt{5}x\right)x\sqrt{5}=20$$

Use the distributive property to multiply -2 by 2\sqrt{5}-\sqrt{5}x.

$$100-50x+\left(-4\sqrt{5}+2\sqrt{5}x\right)x\sqrt{5}=20$$

Use the distributive property to multiply -4\sqrt{5}+2\sqrt{5}x by x.

$$100-50x+\left(-4\sqrt{5}x+2\sqrt{5}x^{2}\right)\sqrt{5}=20$$

Use the distributive property to multiply -4\sqrt{5}x+2\sqrt{5}x^{2} by \sqrt{5}.

$$100-50x-4x\left(\sqrt{5}\right)^{2}+2x^{2}\left(\sqrt{5}\right)^{2}=20$$

The square of \sqrt{5} is 5.

$$100-50x-4x\times 5+2x^{2}\left(\sqrt{5}\right)^{2}=20$$

Multiply -4 and 5 to get -20.

$$100-50x-20x+2x^{2}\left(\sqrt{5}\right)^{2}=20$$

The square of \sqrt{5} is 5.

$$100-50x-20x+2x^{2}\times 5=20$$

Multiply 2 and 5 to get 10.

$$100-50x-20x+10x^{2}=20$$

Combine -50x and -20x to get -70x.

$$100-70x+10x^{2}=20$$

Subtract 100 from both sides.

$$-70x+10x^{2}=20-100$$

Subtract 100 from 20 to get -80.

$$-70x+10x^{2}=-80$$

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.

$$10x^{2}-70x=-80$$

Divide both sides by 10.

$$\frac{10x^{2}-70x}{10}=-\frac{80}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{70}{10}\right)x=-\frac{80}{10}$$

Divide -70 by 10.

$$x^{2}-7x=-\frac{80}{10}$$

Divide -80 by 10.

$$x^{2}-7x=-8$$

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.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-8+\left(-\frac{7}{2}\right)^{2}$$

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

$$x^{2}-7x+\frac{49}{4}=-8+\frac{49}{4}$$

Add -8 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{17}{4}$$

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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{17}}{2} x-\frac{7}{2}=-\frac{\sqrt{17}}{2}$$

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

$$x=\frac{\sqrt{17}+7}{2} x=\frac{7-\sqrt{17}}{2}$$