Solve 80*51=(80+x)(53-x)

$$80 \times 51 = ( 80 + x ) ( 53 - x )$$

Solve for x

$$x=-32$$

$$x=5$$

Steps Using the Quadratic Formula

Multiply 80 and 51 to get 4080.

$$4080=\left(80+x\right)\left(53-x\right)$$

Use the distributive property to multiply 80+x by 53-x and combine like terms.

$$4080=4240-27x-x^{2}$$

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

$$4240-27x-x^{2}=4080$$

Subtract 4080 from both sides.

$$4240-27x-x^{2}-4080=0$$

Subtract 4080 from 4240 to get 160.

$$160-27x-x^{2}=0$$

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.

$$-x^{2}-27x+160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -27 for b, and 160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\left(-1\right)\times 160}}{2\left(-1\right)}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\left(-1\right)\times 160}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-27\right)±\sqrt{729+4\times 160}}{2\left(-1\right)}$$

Multiply 4 times 160.

$$x=\frac{-\left(-27\right)±\sqrt{729+640}}{2\left(-1\right)}$$

Add 729 to 640.

$$x=\frac{-\left(-27\right)±\sqrt{1369}}{2\left(-1\right)}$$

Take the square root of 1369.

$$x=\frac{-\left(-27\right)±37}{2\left(-1\right)}$$

The opposite of -27 is 27.

$$x=\frac{27±37}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{27±37}{-2}$$

Now solve the equation x=\frac{27±37}{-2} when ± is plus. Add 27 to 37.

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

Divide 64 by -2.

$$x=-32$$

Now solve the equation x=\frac{27±37}{-2} when ± is minus. Subtract 37 from 27.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

The equation is now solved.

$$x=-32 x=5$$

Steps for Completing the Square

Multiply 80 and 51 to get 4080.

$$4080=\left(80+x\right)\left(53-x\right)$$

Use the distributive property to multiply 80+x by 53-x and combine like terms.

$$4080=4240-27x-x^{2}$$

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

$$4240-27x-x^{2}=4080$$

Subtract 4240 from both sides.

$$-27x-x^{2}=4080-4240$$

Subtract 4240 from 4080 to get -160.

$$-27x-x^{2}=-160$$

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.

$$-x^{2}-27x=-160$$

Divide both sides by -1.

$$\frac{-x^{2}-27x}{-1}=-\frac{160}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{27}{-1}\right)x=-\frac{160}{-1}$$

Divide -27 by -1.

$$x^{2}+27x=-\frac{160}{-1}$$

Divide -160 by -1.

$$x^{2}+27x=160$$

Divide 27, the coefficient of the x term, by 2 to get \frac{27}{2}. Then add the square of \frac{27}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+27x+\left(\frac{27}{2}\right)^{2}=160+\left(\frac{27}{2}\right)^{2}$$

Square \frac{27}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+27x+\frac{729}{4}=160+\frac{729}{4}$$

Add 160 to \frac{729}{4}.

$$x^{2}+27x+\frac{729}{4}=\frac{1369}{4}$$

Factor x^{2}+27x+\frac{729}{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}.

$$\left(x+\frac{27}{2}\right)^{2}=\frac{1369}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{27}{2}\right)^{2}}=\sqrt{\frac{1369}{4}}$$

Simplify.

$$x+\frac{27}{2}=\frac{37}{2} x+\frac{27}{2}=-\frac{37}{2}$$

Subtract \frac{27}{2} from both sides of the equation.

$$x=5 x=-32$$