Solve (7d+9)(d+7)=47

$$( 7 d + 9 ) ( d + 7 ) = 47$$

Solve for d

$$d=-8$$

$$d=-\frac{2}{7}\approx -0.285714286$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7d+9 by d+7 and combine like terms.

$$7d^{2}+58d+63=47$$

Subtract 47 from both sides.

$$7d^{2}+58d+63-47=0$$

Subtract 47 from 63 to get 16.

$$7d^{2}+58d+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 58 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-58±\sqrt{58^{2}-4\times 7\times 16}}{2\times 7}$$

Square 58.

$$d=\frac{-58±\sqrt{3364-4\times 7\times 16}}{2\times 7}$$

Multiply -4 times 7.

$$d=\frac{-58±\sqrt{3364-28\times 16}}{2\times 7}$$

Multiply -28 times 16.

$$d=\frac{-58±\sqrt{3364-448}}{2\times 7}$$

Add 3364 to -448.

$$d=\frac{-58±\sqrt{2916}}{2\times 7}$$

Take the square root of 2916.

$$d=\frac{-58±54}{2\times 7}$$

Multiply 2 times 7.

$$d=\frac{-58±54}{14}$$

Now solve the equation d=\frac{-58±54}{14} when ± is plus. Add -58 to 54.

$$d=-\frac{4}{14}$$

Reduce the fraction \frac{-4}{14} to lowest terms by extracting and canceling out 2.

$$d=-\frac{2}{7}$$

Now solve the equation d=\frac{-58±54}{14} when ± is minus. Subtract 54 from -58.

$$d=-\frac{112}{14}$$

Divide -112 by 14.

$$d=-8$$

The equation is now solved.

$$d=-\frac{2}{7} d=-8$$

Steps for Completing the Square

Use the distributive property to multiply 7d+9 by d+7 and combine like terms.

$$7d^{2}+58d+63=47$$

Subtract 63 from both sides.

$$7d^{2}+58d=47-63$$

Subtract 63 from 47 to get -16.

$$7d^{2}+58d=-16$$

Divide both sides by 7.

$$\frac{7d^{2}+58d}{7}=-\frac{16}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$d^{2}+\frac{58}{7}d=-\frac{16}{7}$$

Divide \frac{58}{7}, the coefficient of the x term, by 2 to get \frac{29}{7}. Then add the square of \frac{29}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}+\frac{58}{7}d+\left(\frac{29}{7}\right)^{2}=-\frac{16}{7}+\left(\frac{29}{7}\right)^{2}$$

Square \frac{29}{7} by squaring both the numerator and the denominator of the fraction.

$$d^{2}+\frac{58}{7}d+\frac{841}{49}=-\frac{16}{7}+\frac{841}{49}$$

Add -\frac{16}{7} to \frac{841}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$d^{2}+\frac{58}{7}d+\frac{841}{49}=\frac{729}{49}$$

Factor d^{2}+\frac{58}{7}d+\frac{841}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d+\frac{29}{7}\right)^{2}=\frac{729}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+\frac{29}{7}\right)^{2}}=\sqrt{\frac{729}{49}}$$

Simplify.

$$d+\frac{29}{7}=\frac{27}{7} d+\frac{29}{7}=-\frac{27}{7}$$

Subtract \frac{29}{7} from both sides of the equation.

$$d=-\frac{2}{7} d=-8$$