Solve (x-3)(x+4)=17

$$( x - 3 ) ( x + 4 ) = 17$$

Solve for x

$$x = \frac{3 \sqrt{13} - 1}{2} \approx 4.908326913$$

$$x=\frac{-3\sqrt{13}-1}{2}\approx -5.908326913$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-3 by x+4 and combine like terms.

$$x^{2}+x-12=17$$

Subtract 17 from both sides.

$$x^{2}+x-12-17=0$$

Subtract 17 from -12 to get -29.

$$x^{2}+x-29=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -29 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-29\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-29\right)}}{2}$$

Multiply -4 times -29.

$$x=\frac{-1±\sqrt{1+116}}{2}$$

Add 1 to 116.

$$x=\frac{-1±\sqrt{117}}{2}$$

Take the square root of 117.

$$x=\frac{-1±3\sqrt{13}}{2}$$

Now solve the equation x=\frac{-1±3\sqrt{13}}{2} when ± is plus. Add -1 to 3\sqrt{13}.

$$x=\frac{3\sqrt{13}-1}{2}$$

Now solve the equation x=\frac{-1±3\sqrt{13}}{2} when ± is minus. Subtract 3\sqrt{13} from -1.

$$x=\frac{-3\sqrt{13}-1}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{13}-1}{2} x=\frac{-3\sqrt{13}-1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x-3 by x+4 and combine like terms.

$$x^{2}+x-12=17$$

Add 12 to both sides.

$$x^{2}+x=17+12$$

Add 17 and 12 to get 29.

$$x^{2}+x=29$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=29+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=29+\frac{1}{4}$$

Add 29 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{117}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{117}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{117}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{3\sqrt{13}}{2} x+\frac{1}{2}=-\frac{3\sqrt{13}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{3\sqrt{13}-1}{2} x=\frac{-3\sqrt{13}-1}{2}$$