Solve (x+3)(x+8)=42

$$( x + 3 ) ( x + 8 ) = 42$$

Solve for x

$$x = \frac{\sqrt{193} - 11}{2} \approx 1.446221995$$

$$x=\frac{-\sqrt{193}-11}{2}\approx -12.446221995$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+3 by x+8 and combine like terms.

$$x^{2}+11x+24=42$$

Subtract 42 from both sides.

$$x^{2}+11x+24-42=0$$

Subtract 42 from 24 to get -18.

$$x^{2}+11x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 11 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-18\right)}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-11±\sqrt{121+72}}{2}$$

Add 121 to 72.

$$x=\frac{-11±\sqrt{193}}{2}$$

Now solve the equation x=\frac{-11±\sqrt{193}}{2} when ± is plus. Add -11 to \sqrt{193}.

$$x=\frac{\sqrt{193}-11}{2}$$

Now solve the equation x=\frac{-11±\sqrt{193}}{2} when ± is minus. Subtract \sqrt{193} from -11.

$$x=\frac{-\sqrt{193}-11}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{193}-11}{2} x=\frac{-\sqrt{193}-11}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x+3 by x+8 and combine like terms.

$$x^{2}+11x+24=42$$

Subtract 24 from both sides.

$$x^{2}+11x=42-24$$

Subtract 24 from 42 to get 18.

$$x^{2}+11x=18$$

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.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=18+\left(\frac{11}{2}\right)^{2}$$

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

$$x^{2}+11x+\frac{121}{4}=18+\frac{121}{4}$$

Add 18 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{193}{4}$$

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}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{193}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{193}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{\sqrt{193}}{2} x+\frac{11}{2}=-\frac{\sqrt{193}}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=\frac{\sqrt{193}-11}{2} x=\frac{-\sqrt{193}-11}{2}$$