Solve (x+1)(x+3)=5

$$( x + 1 ) ( x + 3 ) = 5$$

Solve for x (complex solution)

$$x=\sqrt{6}-2\approx 0.449489743$$

$$x=-\left(\sqrt{6}+2\right)\approx -4.449489743$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+1 by x+3 and combine like terms.

$$x^{2}+4x+3=5$$

Subtract 5 from both sides.

$$x^{2}+4x+3-5=0$$

Subtract 5 from 3 to get -2.

$$x^{2}+4x-2=0$$

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8}}{2}$$

Add 16 to 8.

$$x=\frac{-4±\sqrt{24}}{2}$$

Take the square root of 24.

$$x=\frac{-4±2\sqrt{6}}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{6}}{2} when ± is plus. Add -4 to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}-4}{2}$$

Divide -4+2\sqrt{6} by 2.

$$x=\sqrt{6}-2$$

Now solve the equation x=\frac{-4±2\sqrt{6}}{2} when ± is minus. Subtract 2\sqrt{6} from -4.

$$x=\frac{-2\sqrt{6}-4}{2}$$

Divide -4-2\sqrt{6} by 2.

$$x=-\sqrt{6}-2$$

The equation is now solved.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$

Steps for Completing the Square

Use the distributive property to multiply x+1 by x+3 and combine like terms.

$$x^{2}+4x+3=5$$

Subtract 3 from both sides.

$$x^{2}+4x=5-3$$

Subtract 3 from 5 to get 2.

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

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

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

Square 2.

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

Add 2 to 4.

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

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{6}$$

Simplify.

$$x+2=\sqrt{6} x+2=-\sqrt{6}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$

Solve for x

$$x=\sqrt{6}-2\approx 0.449489743$$

$$x=-\sqrt{6}-2\approx -4.449489743$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+1 by x+3 and combine like terms.

$$x^{2}+4x+3=5$$

Subtract 5 from both sides.

$$x^{2}+4x+3-5=0$$

Subtract 5 from 3 to get -2.

$$x^{2}+4x-2=0$$

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8}}{2}$$

Add 16 to 8.

$$x=\frac{-4±\sqrt{24}}{2}$$

Take the square root of 24.

$$x=\frac{-4±2\sqrt{6}}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{6}}{2} when ± is plus. Add -4 to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}-4}{2}$$

Divide -4+2\sqrt{6} by 2.

$$x=\sqrt{6}-2$$

Now solve the equation x=\frac{-4±2\sqrt{6}}{2} when ± is minus. Subtract 2\sqrt{6} from -4.

$$x=\frac{-2\sqrt{6}-4}{2}$$

Divide -4-2\sqrt{6} by 2.

$$x=-\sqrt{6}-2$$

The equation is now solved.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$

Steps for Completing the Square

Use the distributive property to multiply x+1 by x+3 and combine like terms.

$$x^{2}+4x+3=5$$

Subtract 3 from both sides.

$$x^{2}+4x=5-3$$

Subtract 3 from 5 to get 2.

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

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

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

Square 2.

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

Add 2 to 4.

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

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{6}$$

Simplify.

$$x+2=\sqrt{6} x+2=-\sqrt{6}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{6}-2 x=-\sqrt{6}-2$$