Solve x(24-4x+4)/2=-520

$$\frac{ x(24-4x+4) }{ 2 } =-520$$

Solve for x

$$x=-13$$

$$x=20$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$x\left(24-4x+4\right)=-520\times 2$$

Add 24 and 4 to get 28.

$$x\left(28-4x\right)=-520\times 2$$

Use the distributive property to multiply x by 28-4x.

$$28x-4x^{2}=-520\times 2$$

Multiply -520 and 2 to get -1040.

$$28x-4x^{2}=-1040$$

Add 1040 to both sides.

$$28x-4x^{2}+1040=0$$

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.

$$-4x^{2}+28x+1040=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 28 for b, and 1040 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-4\right)\times 1040}}{2\left(-4\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-4\right)\times 1040}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-28±\sqrt{784+16\times 1040}}{2\left(-4\right)}$$

Multiply 16 times 1040.

$$x=\frac{-28±\sqrt{784+16640}}{2\left(-4\right)}$$

Add 784 to 16640.

$$x=\frac{-28±\sqrt{17424}}{2\left(-4\right)}$$

Take the square root of 17424.

$$x=\frac{-28±132}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-28±132}{-8}$$

Now solve the equation x=\frac{-28±132}{-8} when ± is plus. Add -28 to 132.

$$x=\frac{104}{-8}$$

Divide 104 by -8.

$$x=-13$$

Now solve the equation x=\frac{-28±132}{-8} when ± is minus. Subtract 132 from -28.

$$x=-\frac{160}{-8}$$

Divide -160 by -8.

$$x=20$$

The equation is now solved.

$$x=-13 x=20$$

Steps for Completing the Square

Multiply both sides by 2.

$$x\left(24-4x+4\right)=-520\times 2$$

Add 24 and 4 to get 28.

$$x\left(28-4x\right)=-520\times 2$$

Use the distributive property to multiply x by 28-4x.

$$28x-4x^{2}=-520\times 2$$

Multiply -520 and 2 to get -1040.

$$28x-4x^{2}=-1040$$

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.

$$-4x^{2}+28x=-1040$$

Divide both sides by -4.

$$\frac{-4x^{2}+28x}{-4}=-\frac{1040}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{28}{-4}x=-\frac{1040}{-4}$$

Divide 28 by -4.

$$x^{2}-7x=-\frac{1040}{-4}$$

Divide -1040 by -4.

$$x^{2}-7x=260$$

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.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=260+\left(-\frac{7}{2}\right)^{2}$$

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

$$x^{2}-7x+\frac{49}{4}=260+\frac{49}{4}$$

Add 260 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{1089}{4}$$

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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{1089}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{1089}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{33}{2} x-\frac{7}{2}=-\frac{33}{2}$$

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

$$x=20 x=-13$$