Solve 4x+2xleft(20-xright)=64

$$4x+2x \left( 20-x \right) =64$$

Solve for x

$$x=\sqrt{89}+11\approx 20.433981132$$

$$x=11-\sqrt{89}\approx 1.566018868$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by 20-x.

$$4x+40x-2x^{2}=64$$

Combine 4x and 40x to get 44x.

$$44x-2x^{2}=64$$

Subtract 64 from both sides.

$$44x-2x^{2}-64=0$$

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.

$$-2x^{2}+44x-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 44 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-44±\sqrt{44^{2}-4\left(-2\right)\left(-64\right)}}{2\left(-2\right)}$$

Square 44.

$$x=\frac{-44±\sqrt{1936-4\left(-2\right)\left(-64\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-44±\sqrt{1936+8\left(-64\right)}}{2\left(-2\right)}$$

Multiply 8 times -64.

$$x=\frac{-44±\sqrt{1936-512}}{2\left(-2\right)}$$

Add 1936 to -512.

$$x=\frac{-44±\sqrt{1424}}{2\left(-2\right)}$$

Take the square root of 1424.

$$x=\frac{-44±4\sqrt{89}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-44±4\sqrt{89}}{-4}$$

Now solve the equation x=\frac{-44±4\sqrt{89}}{-4} when ± is plus. Add -44 to 4\sqrt{89}.

$$x=\frac{4\sqrt{89}-44}{-4}$$

Divide -44+4\sqrt{89} by -4.

$$x=11-\sqrt{89}$$

Now solve the equation x=\frac{-44±4\sqrt{89}}{-4} when ± is minus. Subtract 4\sqrt{89} from -44.

$$x=\frac{-4\sqrt{89}-44}{-4}$$

Divide -44-4\sqrt{89} by -4.

$$x=\sqrt{89}+11$$

The equation is now solved.

$$x=11-\sqrt{89} x=\sqrt{89}+11$$

Steps for Completing the Square

Use the distributive property to multiply 2x by 20-x.

$$4x+40x-2x^{2}=64$$

Combine 4x and 40x to get 44x.

$$44x-2x^{2}=64$$

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.

$$-2x^{2}+44x=64$$

Divide both sides by -2.

$$\frac{-2x^{2}+44x}{-2}=\frac{64}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{44}{-2}x=\frac{64}{-2}$$

Divide 44 by -2.

$$x^{2}-22x=\frac{64}{-2}$$

Divide 64 by -2.

$$x^{2}-22x=-32$$

Divide -22, the coefficient of the x term, by 2 to get -11. Then add the square of -11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-22x+\left(-11\right)^{2}=-32+\left(-11\right)^{2}$$

Square -11.

$$x^{2}-22x+121=-32+121$$

Add -32 to 121.

$$x^{2}-22x+121=89$$

Factor x^{2}-22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-11\right)^{2}=89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-11\right)^{2}}=\sqrt{89}$$

Simplify.

$$x-11=\sqrt{89} x-11=-\sqrt{89}$$

Add 11 to both sides of the equation.

$$x=\sqrt{89}+11 x=11-\sqrt{89}$$