Solve (q+6)(q+2)=21

$$( q + 6 ) ( q + 2 ) = 21$$

Solve for q

$$q=-9$$

$$q=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply q+6 by q+2 and combine like terms.

$$q^{2}+8q+12=21$$

Subtract 21 from both sides.

$$q^{2}+8q+12-21=0$$

Subtract 21 from 12 to get -9.

$$q^{2}+8q-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$q=\frac{-8±\sqrt{8^{2}-4\left(-9\right)}}{2}$$

Square 8.

$$q=\frac{-8±\sqrt{64-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$q=\frac{-8±\sqrt{64+36}}{2}$$

Add 64 to 36.

$$q=\frac{-8±\sqrt{100}}{2}$$

Take the square root of 100.

$$q=\frac{-8±10}{2}$$

Now solve the equation q=\frac{-8±10}{2} when ± is plus. Add -8 to 10.

$$q=\frac{2}{2}$$

Divide 2 by 2.

$$q=1$$

Now solve the equation q=\frac{-8±10}{2} when ± is minus. Subtract 10 from -8.

$$q=-\frac{18}{2}$$

Divide -18 by 2.

$$q=-9$$

The equation is now solved.

$$q=1 q=-9$$

Steps for Completing the Square

Use the distributive property to multiply q+6 by q+2 and combine like terms.

$$q^{2}+8q+12=21$$

Subtract 12 from both sides.

$$q^{2}+8q=21-12$$

Subtract 12 from 21 to get 9.

$$q^{2}+8q=9$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$q^{2}+8q+4^{2}=9+4^{2}$$

Square 4.

$$q^{2}+8q+16=9+16$$

Add 9 to 16.

$$q^{2}+8q+16=25$$

Factor q^{2}+8q+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}.

$$\left(q+4\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(q+4\right)^{2}}=\sqrt{25}$$

Simplify.

$$q+4=5 q+4=-5$$

Subtract 4 from both sides of the equation.

$$q=1 q=-9$$