Solve (x+5)(x+3)=40

$$( x + 5 ) ( x + 3 ) = 40$$

Solve for x (complex solution)

$$x=\sqrt{41}-4\approx 2.403124237$$

$$x=-\left(\sqrt{41}+4\right)\approx -10.403124237$$

Steps Using the Quadratic Formula

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

$$x^{2}+8x+15=40$$

Subtract 40 from both sides.

$$x^{2}+8x+15-40=0$$

Subtract 40 from 15 to get -25.

$$x^{2}+8x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-25\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$x=\frac{-8±\sqrt{64+100}}{2}$$

Add 64 to 100.

$$x=\frac{-8±\sqrt{164}}{2}$$

Take the square root of 164.

$$x=\frac{-8±2\sqrt{41}}{2}$$

Now solve the equation x=\frac{-8±2\sqrt{41}}{2} when ± is plus. Add -8 to 2\sqrt{41}.

$$x=\frac{2\sqrt{41}-8}{2}$$

Divide -8+2\sqrt{41} by 2.

$$x=\sqrt{41}-4$$

Now solve the equation x=\frac{-8±2\sqrt{41}}{2} when ± is minus. Subtract 2\sqrt{41} from -8.

$$x=\frac{-2\sqrt{41}-8}{2}$$

Divide -8-2\sqrt{41} by 2.

$$x=-\sqrt{41}-4$$

The equation is now solved.

$$x=\sqrt{41}-4 x=-\sqrt{41}-4$$

Steps for Completing the Square

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

$$x^{2}+8x+15=40$$

Subtract 15 from both sides.

$$x^{2}+8x=40-15$$

Subtract 15 from 40 to get 25.

$$x^{2}+8x=25$$

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.

$$x^{2}+8x+4^{2}=25+4^{2}$$

Square 4.

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

Add 25 to 16.

$$x^{2}+8x+16=41$$

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}.

$$\left(x+4\right)^{2}=41$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{41}$$

Simplify.

$$x+4=\sqrt{41} x+4=-\sqrt{41}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{41}-4 x=-\sqrt{41}-4$$

Solve for x

$$x=\sqrt{41}-4\approx 2.403124237$$

$$x=-\sqrt{41}-4\approx -10.403124237$$

Steps Using the Quadratic Formula

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

$$x^{2}+8x+15=40$$

Subtract 40 from both sides.

$$x^{2}+8x+15-40=0$$

Subtract 40 from 15 to get -25.

$$x^{2}+8x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-25\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$x=\frac{-8±\sqrt{64+100}}{2}$$

Add 64 to 100.

$$x=\frac{-8±\sqrt{164}}{2}$$

Take the square root of 164.

$$x=\frac{-8±2\sqrt{41}}{2}$$

Now solve the equation x=\frac{-8±2\sqrt{41}}{2} when ± is plus. Add -8 to 2\sqrt{41}.

$$x=\frac{2\sqrt{41}-8}{2}$$

Divide -8+2\sqrt{41} by 2.

$$x=\sqrt{41}-4$$

Now solve the equation x=\frac{-8±2\sqrt{41}}{2} when ± is minus. Subtract 2\sqrt{41} from -8.

$$x=\frac{-2\sqrt{41}-8}{2}$$

Divide -8-2\sqrt{41} by 2.

$$x=-\sqrt{41}-4$$

The equation is now solved.

$$x=\sqrt{41}-4 x=-\sqrt{41}-4$$

Steps for Completing the Square

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

$$x^{2}+8x+15=40$$

Subtract 15 from both sides.

$$x^{2}+8x=40-15$$

Subtract 15 from 40 to get 25.

$$x^{2}+8x=25$$

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.

$$x^{2}+8x+4^{2}=25+4^{2}$$

Square 4.

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

Add 25 to 16.

$$x^{2}+8x+16=41$$

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}.

$$\left(x+4\right)^{2}=41$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{41}$$

Simplify.

$$x+4=\sqrt{41} x+4=-\sqrt{41}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{41}-4 x=-\sqrt{41}-4$$