Solve x(160-2x)-30x-500=1300

$$x ( 160 - 2 x ) - 30 x - 500 = 1300$$

Solve for x

$$x=20$$

$$x=45$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 160-2x.

$$160x-2x^{2}-30x-500=1300$$

Combine 160x and -30x to get 130x.

$$130x-2x^{2}-500=1300$$

Subtract 1300 from both sides.

$$130x-2x^{2}-500-1300=0$$

Subtract 1300 from -500 to get -1800.

$$130x-2x^{2}-1800=0$$

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.

$$-2x^{2}+130x-1800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 130 for b, and -1800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-130±\sqrt{130^{2}-4\left(-2\right)\left(-1800\right)}}{2\left(-2\right)}$$

Square 130.

$$x=\frac{-130±\sqrt{16900-4\left(-2\right)\left(-1800\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-130±\sqrt{16900+8\left(-1800\right)}}{2\left(-2\right)}$$

Multiply 8 times -1800.

$$x=\frac{-130±\sqrt{16900-14400}}{2\left(-2\right)}$$

Add 16900 to -14400.

$$x=\frac{-130±\sqrt{2500}}{2\left(-2\right)}$$

Take the square root of 2500.

$$x=\frac{-130±50}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-130±50}{-4}$$

Now solve the equation x=\frac{-130±50}{-4} when ± is plus. Add -130 to 50.

$$x=-\frac{80}{-4}$$

Divide -80 by -4.

$$x=20$$

Now solve the equation x=\frac{-130±50}{-4} when ± is minus. Subtract 50 from -130.

$$x=-\frac{180}{-4}$$

Divide -180 by -4.

$$x=45$$

The equation is now solved.

$$x=20 x=45$$

Steps for Completing the Square

Use the distributive property to multiply x by 160-2x.

$$160x-2x^{2}-30x-500=1300$$

Combine 160x and -30x to get 130x.

$$130x-2x^{2}-500=1300$$

Add 500 to both sides.

$$130x-2x^{2}=1300+500$$

Add 1300 and 500 to get 1800.

$$130x-2x^{2}=1800$$

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.

$$-2x^{2}+130x=1800$$

Divide both sides by -2.

$$\frac{-2x^{2}+130x}{-2}=\frac{1800}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{130}{-2}x=\frac{1800}{-2}$$

Divide 130 by -2.

$$x^{2}-65x=\frac{1800}{-2}$$

Divide 1800 by -2.

$$x^{2}-65x=-900$$

Divide -65, the coefficient of the x term, by 2 to get -\frac{65}{2}. Then add the square of -\frac{65}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-65x+\left(-\frac{65}{2}\right)^{2}=-900+\left(-\frac{65}{2}\right)^{2}$$

Square -\frac{65}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-65x+\frac{4225}{4}=-900+\frac{4225}{4}$$

Add -900 to \frac{4225}{4}.

$$x^{2}-65x+\frac{4225}{4}=\frac{625}{4}$$

Factor x^{2}-65x+\frac{4225}{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}.

$$\left(x-\frac{65}{2}\right)^{2}=\frac{625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{65}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Simplify.

$$x-\frac{65}{2}=\frac{25}{2} x-\frac{65}{2}=-\frac{25}{2}$$

Add \frac{65}{2} to both sides of the equation.

$$x=45 x=20$$