Solve (x+00)(1300-5x)=32000

$$( x + 00 ) ( 1300 - 5 x ) = 32000$$

Solve for x

$$x=10\sqrt{105}+130\approx 232.46950766$$

$$x=130-10\sqrt{105}\approx 27.53049234$$

Steps Using the Quadratic Formula

Multiply 0 and 0 to get 0.

$$\left(x+0\right)\left(1300-5x\right)=32000$$

Anything plus zero gives itself.

$$x\left(1300-5x\right)=32000$$

Use the distributive property to multiply x by 1300-5x.

$$1300x-5x^{2}=32000$$

Subtract 32000 from both sides.

$$1300x-5x^{2}-32000=0$$

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.

$$-5x^{2}+1300x-32000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 1300 for b, and -32000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1300±\sqrt{1300^{2}-4\left(-5\right)\left(-32000\right)}}{2\left(-5\right)}$$

Square 1300.

$$x=\frac{-1300±\sqrt{1690000-4\left(-5\right)\left(-32000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-1300±\sqrt{1690000+20\left(-32000\right)}}{2\left(-5\right)}$$

Multiply 20 times -32000.

$$x=\frac{-1300±\sqrt{1690000-640000}}{2\left(-5\right)}$$

Add 1690000 to -640000.

$$x=\frac{-1300±\sqrt{1050000}}{2\left(-5\right)}$$

Take the square root of 1050000.

$$x=\frac{-1300±100\sqrt{105}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-1300±100\sqrt{105}}{-10}$$

Now solve the equation x=\frac{-1300±100\sqrt{105}}{-10} when ± is plus. Add -1300 to 100\sqrt{105}.

$$x=\frac{100\sqrt{105}-1300}{-10}$$

Divide -1300+100\sqrt{105} by -10.

$$x=130-10\sqrt{105}$$

Now solve the equation x=\frac{-1300±100\sqrt{105}}{-10} when ± is minus. Subtract 100\sqrt{105} from -1300.

$$x=\frac{-100\sqrt{105}-1300}{-10}$$

Divide -1300-100\sqrt{105} by -10.

$$x=10\sqrt{105}+130$$

The equation is now solved.

$$x=130-10\sqrt{105} x=10\sqrt{105}+130$$

Steps for Completing the Square

Multiply 0 and 0 to get 0.

$$\left(x+0\right)\left(1300-5x\right)=32000$$

Anything plus zero gives itself.

$$x\left(1300-5x\right)=32000$$

Use the distributive property to multiply x by 1300-5x.

$$1300x-5x^{2}=32000$$

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.

$$-5x^{2}+1300x=32000$$

Divide both sides by -5.

$$\frac{-5x^{2}+1300x}{-5}=\frac{32000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{1300}{-5}x=\frac{32000}{-5}$$

Divide 1300 by -5.

$$x^{2}-260x=\frac{32000}{-5}$$

Divide 32000 by -5.

$$x^{2}-260x=-6400$$

Divide -260, the coefficient of the x term, by 2 to get -130. Then add the square of -130 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-260x+\left(-130\right)^{2}=-6400+\left(-130\right)^{2}$$

Square -130.

$$x^{2}-260x+16900=-6400+16900$$

Add -6400 to 16900.

$$x^{2}-260x+16900=10500$$

Factor x^{2}-260x+16900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-130\right)^{2}=10500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-130\right)^{2}}=\sqrt{10500}$$

Simplify.

$$x-130=10\sqrt{105} x-130=-10\sqrt{105}$$

Add 130 to both sides of the equation.

$$x=10\sqrt{105}+130 x=130-10\sqrt{105}$$