Solve (x-6)(x-5)+(x-7)(x-4)=10

$$( x - 6 ) ( x - 5 ) + ( x - 7 ) ( x - 4 ) = 10$$

Solve for x

$$x=3$$

$$x=8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-6 by x-5 and combine like terms.

$$x^{2}-11x+30+\left(x-7\right)\left(x-4\right)=10$$

Use the distributive property to multiply x-7 by x-4 and combine like terms.

$$x^{2}-11x+30+x^{2}-11x+28=10$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-11x+30-11x+28=10$$

Combine -11x and -11x to get -22x.

$$2x^{2}-22x+30+28=10$$

Add 30 and 28 to get 58.

$$2x^{2}-22x+58=10$$

Subtract 10 from both sides.

$$2x^{2}-22x+58-10=0$$

Subtract 10 from 58 to get 48.

$$2x^{2}-22x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -22 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 2\times 48}}{2\times 2}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 2\times 48}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-22\right)±\sqrt{484-8\times 48}}{2\times 2}$$

Multiply -8 times 48.

$$x=\frac{-\left(-22\right)±\sqrt{484-384}}{2\times 2}$$

Add 484 to -384.

$$x=\frac{-\left(-22\right)±\sqrt{100}}{2\times 2}$$

Take the square root of 100.

$$x=\frac{-\left(-22\right)±10}{2\times 2}$$

The opposite of -22 is 22.

$$x=\frac{22±10}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{22±10}{4}$$

Now solve the equation x=\frac{22±10}{4} when ± is plus. Add 22 to 10.

$$x=\frac{32}{4}$$

Divide 32 by 4.

$$x=8$$

Now solve the equation x=\frac{22±10}{4} when ± is minus. Subtract 10 from 22.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

The equation is now solved.

$$x=8 x=3$$

Steps for Completing the Square

Use the distributive property to multiply x-6 by x-5 and combine like terms.

$$x^{2}-11x+30+\left(x-7\right)\left(x-4\right)=10$$

Use the distributive property to multiply x-7 by x-4 and combine like terms.

$$x^{2}-11x+30+x^{2}-11x+28=10$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-11x+30-11x+28=10$$

Combine -11x and -11x to get -22x.

$$2x^{2}-22x+30+28=10$$

Add 30 and 28 to get 58.

$$2x^{2}-22x+58=10$$

Subtract 58 from both sides.

$$2x^{2}-22x=10-58$$

Subtract 58 from 10 to get -48.

$$2x^{2}-22x=-48$$

Divide both sides by 2.

$$\frac{2x^{2}-22x}{2}=-\frac{48}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{22}{2}\right)x=-\frac{48}{2}$$

Divide -22 by 2.

$$x^{2}-11x=-\frac{48}{2}$$

Divide -48 by 2.

$$x^{2}-11x=-24$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-24+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-24+\frac{121}{4}$$

Add -24 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{25}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{5}{2} x-\frac{11}{2}=-\frac{5}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=8 x=3$$