Solve 10/7*4*9=(20-x)x

$$\frac { 10 } { 7 } \times 4 \times 9 = ( 20 - x ) x$$

Solve for x

$$x=\frac{2\sqrt{595}}{7}+10\approx 16.969320524$$

$$x=-\frac{2\sqrt{595}}{7}+10\approx 3.030679476$$

Steps Using the Quadratic Formula

Express \frac{10}{7}\times 4 as a single fraction.

$$\frac{10\times 4}{7}\times 9=\left(20-x\right)x$$

Multiply 10 and 4 to get 40.

$$\frac{40}{7}\times 9=\left(20-x\right)x$$

Express \frac{40}{7}\times 9 as a single fraction.

$$\frac{40\times 9}{7}=\left(20-x\right)x$$

Multiply 40 and 9 to get 360.

$$\frac{360}{7}=\left(20-x\right)x$$

Use the distributive property to multiply 20-x by x.

$$\frac{360}{7}=20x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$20x-x^{2}=\frac{360}{7}$$

Subtract \frac{360}{7} from both sides.

$$20x-x^{2}-\frac{360}{7}=0$$

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.

$$-x^{2}+20x-\frac{360}{7}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 20 for b, and -\frac{360}{7} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-1\right)\left(-\frac{360}{7}\right)}}{2\left(-1\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-1\right)\left(-\frac{360}{7}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-20±\sqrt{400+4\left(-\frac{360}{7}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{360}{7}.

$$x=\frac{-20±\sqrt{400-\frac{1440}{7}}}{2\left(-1\right)}$$

Add 400 to -\frac{1440}{7}.

$$x=\frac{-20±\sqrt{\frac{1360}{7}}}{2\left(-1\right)}$$

Take the square root of \frac{1360}{7}.

$$x=\frac{-20±\frac{4\sqrt{595}}{7}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-20±\frac{4\sqrt{595}}{7}}{-2}$$

Now solve the equation x=\frac{-20±\frac{4\sqrt{595}}{7}}{-2} when ± is plus. Add -20 to \frac{4\sqrt{595}}{7}.

$$x=\frac{\frac{4\sqrt{595}}{7}-20}{-2}$$

Divide -20+\frac{4\sqrt{595}}{7} by -2.

$$x=-\frac{2\sqrt{595}}{7}+10$$

Now solve the equation x=\frac{-20±\frac{4\sqrt{595}}{7}}{-2} when ± is minus. Subtract \frac{4\sqrt{595}}{7} from -20.

$$x=\frac{-\frac{4\sqrt{595}}{7}-20}{-2}$$

Divide -20-\frac{4\sqrt{595}}{7} by -2.

$$x=\frac{2\sqrt{595}}{7}+10$$

The equation is now solved.

$$x=-\frac{2\sqrt{595}}{7}+10 x=\frac{2\sqrt{595}}{7}+10$$

Steps for Completing the Square

Express \frac{10}{7}\times 4 as a single fraction.

$$\frac{10\times 4}{7}\times 9=\left(20-x\right)x$$

Multiply 10 and 4 to get 40.

$$\frac{40}{7}\times 9=\left(20-x\right)x$$

Express \frac{40}{7}\times 9 as a single fraction.

$$\frac{40\times 9}{7}=\left(20-x\right)x$$

Multiply 40 and 9 to get 360.

$$\frac{360}{7}=\left(20-x\right)x$$

Use the distributive property to multiply 20-x by x.

$$\frac{360}{7}=20x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$20x-x^{2}=\frac{360}{7}$$

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.

$$-x^{2}+20x=\frac{360}{7}$$

Divide both sides by -1.

$$\frac{-x^{2}+20x}{-1}=\frac{\frac{360}{7}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{20}{-1}x=\frac{\frac{360}{7}}{-1}$$

Divide 20 by -1.

$$x^{2}-20x=\frac{\frac{360}{7}}{-1}$$

Divide \frac{360}{7} by -1.

$$x^{2}-20x=-\frac{360}{7}$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-\frac{360}{7}+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-\frac{360}{7}+100$$

Add -\frac{360}{7} to 100.

$$x^{2}-20x+100=\frac{340}{7}$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=\frac{340}{7}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{\frac{340}{7}}$$

Simplify.

$$x-10=\frac{2\sqrt{595}}{7} x-10=-\frac{2\sqrt{595}}{7}$$

Add 10 to both sides of the equation.

$$x=\frac{2\sqrt{595}}{7}+10 x=-\frac{2\sqrt{595}}{7}+10$$