Solve (2x-17)(x-5)-(3x+1)*(x-7)=84

$$(2x-17)(x-5)-(3x+1) \times (x-7)=84$$

Solve for x

$$x=-8$$

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-17 by x-5 and combine like terms.

$$2x^{2}-27x+85-\left(3x+1\right)\left(x-7\right)=84$$

Use the distributive property to multiply 3x+1 by x-7 and combine like terms.

$$2x^{2}-27x+85-\left(3x^{2}-20x-7\right)=84$$

To find the opposite of 3x^{2}-20x-7, find the opposite of each term.

$$2x^{2}-27x+85-3x^{2}+20x+7=84$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-27x+85+20x+7=84$$

Combine -27x and 20x to get -7x.

$$-x^{2}-7x+85+7=84$$

Add 85 and 7 to get 92.

$$-x^{2}-7x+92=84$$

Subtract 84 from both sides.

$$-x^{2}-7x+92-84=0$$

Subtract 84 from 92 to get 8.

$$-x^{2}-7x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -7 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-7\right)±\sqrt{49+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$x=\frac{-\left(-7\right)±\sqrt{49+32}}{2\left(-1\right)}$$

Add 49 to 32.

$$x=\frac{-\left(-7\right)±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$x=\frac{-\left(-7\right)±9}{2\left(-1\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{7±9}{-2}$$

Now solve the equation x=\frac{7±9}{-2} when ± is plus. Add 7 to 9.

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{7±9}{-2} when ± is minus. Subtract 9 from 7.

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

Divide -2 by -2.

$$x=1$$

The equation is now solved.

$$x=-8 x=1$$

Steps for Completing the Square

Use the distributive property to multiply 2x-17 by x-5 and combine like terms.

$$2x^{2}-27x+85-\left(3x+1\right)\left(x-7\right)=84$$

Use the distributive property to multiply 3x+1 by x-7 and combine like terms.

$$2x^{2}-27x+85-\left(3x^{2}-20x-7\right)=84$$

To find the opposite of 3x^{2}-20x-7, find the opposite of each term.

$$2x^{2}-27x+85-3x^{2}+20x+7=84$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-27x+85+20x+7=84$$

Combine -27x and 20x to get -7x.

$$-x^{2}-7x+85+7=84$$

Add 85 and 7 to get 92.

$$-x^{2}-7x+92=84$$

Subtract 92 from both sides.

$$-x^{2}-7x=84-92$$

Subtract 92 from 84 to get -8.

$$-x^{2}-7x=-8$$

Divide both sides by -1.

$$\frac{-x^{2}-7x}{-1}=-\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{7}{-1}\right)x=-\frac{8}{-1}$$

Divide -7 by -1.

$$x^{2}+7x=-\frac{8}{-1}$$

Divide -8 by -1.

$$x^{2}+7x=8$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=8+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=8+\frac{49}{4}$$

Add 8 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{81}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{9}{2} x+\frac{7}{2}=-\frac{9}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=1 x=-8$$