Solve (12-x)(8-x)=77

$$( 12 - x ) ( 8 - x ) = 77$$

Solve for x

$$x=19$$

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12-x by 8-x and combine like terms.

$$96-20x+x^{2}=77$$

Subtract 77 from both sides.

$$96-20x+x^{2}-77=0$$

Subtract 77 from 96 to get 19.

$$19-20x+x^{2}=0$$

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.

$$x^{2}-20x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 19}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 19}}{2}$$

Multiply -4 times 19.

$$x=\frac{-\left(-20\right)±\sqrt{400-76}}{2}$$

Add 400 to -76.

$$x=\frac{-\left(-20\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-\left(-20\right)±18}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±18}{2}$$

Now solve the equation x=\frac{20±18}{2} when ± is plus. Add 20 to 18.

$$x=\frac{38}{2}$$

Divide 38 by 2.

$$x=19$$

Now solve the equation x=\frac{20±18}{2} when ± is minus. Subtract 18 from 20.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=19 x=1$$

Steps for Completing the Square

Use the distributive property to multiply 12-x by 8-x and combine like terms.

$$96-20x+x^{2}=77$$

Subtract 96 from both sides.

$$-20x+x^{2}=77-96$$

Subtract 96 from 77 to get -19.

$$-20x+x^{2}=-19$$

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.

$$x^{2}-20x=-19$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-19+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-19+100$$

Add -19 to 100.

$$x^{2}-20x+100=81$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-10=9 x-10=-9$$

Add 10 to both sides of the equation.

$$x=19 x=1$$