Solve 25=(x+4)(x)

$$25=(x+4)(x)$$

Solve for x (complex solution)

$$x=\sqrt{29}-2\approx 3.385164807$$

$$x=-\left(\sqrt{29}+2\right)\approx -7.385164807$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+4 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 25 from both sides.

$$x^{2}+4x-25=0$$

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$x=\frac{-4±\sqrt{16+100}}{2}$$

Add 16 to 100.

$$x=\frac{-4±\sqrt{116}}{2}$$

Take the square root of 116.

$$x=\frac{-4±2\sqrt{29}}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{29}}{2} when ± is plus. Add -4 to 2\sqrt{29}.

$$x=\frac{2\sqrt{29}-4}{2}$$

Divide -4+2\sqrt{29} by 2.

$$x=\sqrt{29}-2$$

Now solve the equation x=\frac{-4±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from -4.

$$x=\frac{-2\sqrt{29}-4}{2}$$

Divide -4-2\sqrt{29} by 2.

$$x=-\sqrt{29}-2$$

The equation is now solved.

$$x=\sqrt{29}-2 x=-\sqrt{29}-2$$

Steps for Completing the Square

Use the distributive property to multiply x+4 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

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.

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

Square 2.

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

Add 25 to 4.

$$x^{2}+4x+4=29$$

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}.

$$\left(x+2\right)^{2}=29$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{29}$$

Simplify.

$$x+2=\sqrt{29} x+2=-\sqrt{29}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{29}-2 x=-\sqrt{29}-2$$

Solve for x

$$x=\sqrt{29}-2\approx 3.385164807$$

$$x=-\sqrt{29}-2\approx -7.385164807$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+4 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 25 from both sides.

$$x^{2}+4x-25=0$$

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$x=\frac{-4±\sqrt{16+100}}{2}$$

Add 16 to 100.

$$x=\frac{-4±\sqrt{116}}{2}$$

Take the square root of 116.

$$x=\frac{-4±2\sqrt{29}}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{29}}{2} when ± is plus. Add -4 to 2\sqrt{29}.

$$x=\frac{2\sqrt{29}-4}{2}$$

Divide -4+2\sqrt{29} by 2.

$$x=\sqrt{29}-2$$

Now solve the equation x=\frac{-4±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from -4.

$$x=\frac{-2\sqrt{29}-4}{2}$$

Divide -4-2\sqrt{29} by 2.

$$x=-\sqrt{29}-2$$

The equation is now solved.

$$x=\sqrt{29}-2 x=-\sqrt{29}-2$$

Steps for Completing the Square

Use the distributive property to multiply x+4 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

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.

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

Square 2.

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

Add 25 to 4.

$$x^{2}+4x+4=29$$

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}.

$$\left(x+2\right)^{2}=29$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{29}$$

Simplify.

$$x+2=\sqrt{29} x+2=-\sqrt{29}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{29}-2 x=-\sqrt{29}-2$$