Solve 75*18=(75+x)(18-x)

$$75 \times 18 = ( 75 + x ) ( 18 - x )$$

Solve for x

$$x=-57$$

$$x=0$$

Steps Using the Quadratic Formula

Multiply 75 and 18 to get 1350.

$$1350=\left(75+x\right)\left(18-x\right)$$

Use the distributive property to multiply 75+x by 18-x and combine like terms.

$$1350=1350-57x-x^{2}$$

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

$$1350-57x-x^{2}=1350$$

Subtract 1350 from both sides.

$$1350-57x-x^{2}-1350=0$$

Subtract 1350 from 1350 to get 0.

$$-57x-x^{2}=0$$

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.

$$-x^{2}-57x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -57 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-57\right)±\sqrt{\left(-57\right)^{2}}}{2\left(-1\right)}$$

Take the square root of \left(-57\right)^{2}.

$$x=\frac{-\left(-57\right)±57}{2\left(-1\right)}$$

The opposite of -57 is 57.

$$x=\frac{57±57}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{57±57}{-2}$$

Now solve the equation x=\frac{57±57}{-2} when ± is plus. Add 57 to 57.

$$x=\frac{114}{-2}$$

Divide 114 by -2.

$$x=-57$$

Now solve the equation x=\frac{57±57}{-2} when ± is minus. Subtract 57 from 57.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

The equation is now solved.

$$x=-57 x=0$$

Steps for Completing the Square

Multiply 75 and 18 to get 1350.

$$1350=\left(75+x\right)\left(18-x\right)$$

Use the distributive property to multiply 75+x by 18-x and combine like terms.

$$1350=1350-57x-x^{2}$$

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

$$1350-57x-x^{2}=1350$$

Subtract 1350 from both sides.

$$-57x-x^{2}=1350-1350$$

Subtract 1350 from 1350 to get 0.

$$-57x-x^{2}=0$$

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.

$$-x^{2}-57x=0$$

Divide both sides by -1.

$$\frac{-x^{2}-57x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{57}{-1}\right)x=\frac{0}{-1}$$

Divide -57 by -1.

$$x^{2}+57x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}+57x=0$$

Divide 57, the coefficient of the x term, by 2 to get \frac{57}{2}. Then add the square of \frac{57}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+57x+\left(\frac{57}{2}\right)^{2}=\left(\frac{57}{2}\right)^{2}$$

Square \frac{57}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+57x+\frac{3249}{4}=\frac{3249}{4}$$

Factor x^{2}+57x+\frac{3249}{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}.

$$\left(x+\frac{57}{2}\right)^{2}=\frac{3249}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{57}{2}\right)^{2}}=\sqrt{\frac{3249}{4}}$$

Simplify.

$$x+\frac{57}{2}=\frac{57}{2} x+\frac{57}{2}=-\frac{57}{2}$$

Subtract \frac{57}{2} from both sides of the equation.

$$x=0 x=-57$$