Solve (2x-7)(x+1)-6quad2x-7=0

$$( 2 x - 7 ) ( x + 1 ) - 6 \quad 2 x - 7 = 0$$

Solve for x

$$x = \frac{\sqrt{401} + 17}{4} \approx 9.256246099$$

$$x=\frac{17-\sqrt{401}}{4}\approx -0.756246099$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-7 by x+1 and combine like terms.

$$2x^{2}-5x-7-6\times 2x-7=0$$

Multiply 6 and 2 to get 12.

$$2x^{2}-5x-7-12x-7=0$$

Combine -5x and -12x to get -17x.

$$2x^{2}-17x-7-7=0$$

Subtract 7 from -7 to get -14.

$$2x^{2}-17x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -17 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 2\left(-14\right)}}{2\times 2}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 2\left(-14\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-17\right)±\sqrt{289-8\left(-14\right)}}{2\times 2}$$

Multiply -8 times -14.

$$x=\frac{-\left(-17\right)±\sqrt{289+112}}{2\times 2}$$

Add 289 to 112.

$$x=\frac{-\left(-17\right)±\sqrt{401}}{2\times 2}$$

The opposite of -17 is 17.

$$x=\frac{17±\sqrt{401}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{17±\sqrt{401}}{4}$$

Now solve the equation x=\frac{17±\sqrt{401}}{4} when ± is plus. Add 17 to \sqrt{401}.

$$x=\frac{\sqrt{401}+17}{4}$$

Now solve the equation x=\frac{17±\sqrt{401}}{4} when ± is minus. Subtract \sqrt{401} from 17.

$$x=\frac{17-\sqrt{401}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{401}+17}{4} x=\frac{17-\sqrt{401}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 2x-7 by x+1 and combine like terms.

$$2x^{2}-5x-7-6\times 2x-7=0$$

Multiply 6 and 2 to get 12.

$$2x^{2}-5x-7-12x-7=0$$

Combine -5x and -12x to get -17x.

$$2x^{2}-17x-7-7=0$$

Subtract 7 from -7 to get -14.

$$2x^{2}-17x-14=0$$

Add 14 to both sides. Anything plus zero gives itself.

$$2x^{2}-17x=14$$

Divide both sides by 2.

$$\frac{2x^{2}-17x}{2}=\frac{14}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{17}{2}x=\frac{14}{2}$$

Divide 14 by 2.

$$x^{2}-\frac{17}{2}x=7$$

Divide -\frac{17}{2}, the coefficient of the x term, by 2 to get -\frac{17}{4}. Then add the square of -\frac{17}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{2}x+\left(-\frac{17}{4}\right)^{2}=7+\left(-\frac{17}{4}\right)^{2}$$

Square -\frac{17}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{2}x+\frac{289}{16}=7+\frac{289}{16}$$

Add 7 to \frac{289}{16}.

$$x^{2}-\frac{17}{2}x+\frac{289}{16}=\frac{401}{16}$$

Factor x^{2}-\frac{17}{2}x+\frac{289}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{4}\right)^{2}=\frac{401}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{4}\right)^{2}}=\sqrt{\frac{401}{16}}$$

Simplify.

$$x-\frac{17}{4}=\frac{\sqrt{401}}{4} x-\frac{17}{4}=-\frac{\sqrt{401}}{4}$$

Add \frac{17}{4} to both sides of the equation.

$$x=\frac{\sqrt{401}+17}{4} x=\frac{17-\sqrt{401}}{4}$$