Solve (x+3)(x+5)=9

$$( x + 3 ) ( x + 5 ) = 9$$

Solve for x (complex solution)

$$x=\sqrt{10}-4\approx -0.83772234$$

$$x=-\left(\sqrt{10}+4\right)\approx -7.16227766$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+3 by x+5 and combine like terms.

$$x^{2}+8x+15=9$$

Subtract 9 from both sides.

$$x^{2}+8x+15-9=0$$

Subtract 9 from 15 to get 6.

$$x^{2}+8x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 6}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-8±\sqrt{64-24}}{2}$$

Add 64 to -24.

$$x=\frac{-8±\sqrt{40}}{2}$$

Take the square root of 40.

$$x=\frac{-8±2\sqrt{10}}{2}$$

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

$$x=\frac{2\sqrt{10}-8}{2}$$

Divide -8+2\sqrt{10} by 2.

$$x=\sqrt{10}-4$$

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

$$x=\frac{-2\sqrt{10}-8}{2}$$

Divide -8-2\sqrt{10} by 2.

$$x=-\sqrt{10}-4$$

The equation is now solved.

$$x=\sqrt{10}-4 x=-\sqrt{10}-4$$

Steps for Completing the Square

Use the distributive property to multiply x+3 by x+5 and combine like terms.

$$x^{2}+8x+15=9$$

Subtract 15 from both sides.

$$x^{2}+8x=9-15$$

Subtract 15 from 9 to get -6.

$$x^{2}+8x=-6$$

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.

$$x^{2}+8x+4^{2}=-6+4^{2}$$

Square 4.

$$x^{2}+8x+16=-6+16$$

Add -6 to 16.

$$x^{2}+8x+16=10$$

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

$$\left(x+4\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{10}$$

Simplify.

$$x+4=\sqrt{10} x+4=-\sqrt{10}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{10}-4 x=-\sqrt{10}-4$$

Solve for x

$$x=\sqrt{10}-4\approx -0.83772234$$

$$x=-\sqrt{10}-4\approx -7.16227766$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+3 by x+5 and combine like terms.

$$x^{2}+8x+15=9$$

Subtract 9 from both sides.

$$x^{2}+8x+15-9=0$$

Subtract 9 from 15 to get 6.

$$x^{2}+8x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 6}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-8±\sqrt{64-24}}{2}$$

Add 64 to -24.

$$x=\frac{-8±\sqrt{40}}{2}$$

Take the square root of 40.

$$x=\frac{-8±2\sqrt{10}}{2}$$

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

$$x=\frac{2\sqrt{10}-8}{2}$$

Divide -8+2\sqrt{10} by 2.

$$x=\sqrt{10}-4$$

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

$$x=\frac{-2\sqrt{10}-8}{2}$$

Divide -8-2\sqrt{10} by 2.

$$x=-\sqrt{10}-4$$

The equation is now solved.

$$x=\sqrt{10}-4 x=-\sqrt{10}-4$$

Steps for Completing the Square

Use the distributive property to multiply x+3 by x+5 and combine like terms.

$$x^{2}+8x+15=9$$

Subtract 15 from both sides.

$$x^{2}+8x=9-15$$

Subtract 15 from 9 to get -6.

$$x^{2}+8x=-6$$

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.

$$x^{2}+8x+4^{2}=-6+4^{2}$$

Square 4.

$$x^{2}+8x+16=-6+16$$

Add -6 to 16.

$$x^{2}+8x+16=10$$

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

$$\left(x+4\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{10}$$

Simplify.

$$x+4=\sqrt{10} x+4=-\sqrt{10}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{10}-4 x=-\sqrt{10}-4$$