Solve 75=(x+2)^2,w

$$75 = ( x + 2 ) ^ { 2 } , w$$

Solve for x

$$x=5\sqrt{3}-2\approx 6.660254038$$

$$x=-5\sqrt{3}-2\approx -10.660254038$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$75=x^{2}+4x+4$$

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

$$x^{2}+4x+4=75$$

Subtract 75 from both sides.

$$x^{2}+4x+4-75=0$$

Subtract 75 from 4 to get -71.

$$x^{2}+4x-71=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -71 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-71\right)}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-71\right)}}{2}$$

Multiply -4 times -71.

$$x=\frac{-4±\sqrt{16+284}}{2}$$

Add 16 to 284.

$$x=\frac{-4±\sqrt{300}}{2}$$

Take the square root of 300.

$$x=\frac{-4±10\sqrt{3}}{2}$$

Now solve the equation x=\frac{-4±10\sqrt{3}}{2} when ± is plus. Add -4 to 10\sqrt{3}.

$$x=\frac{10\sqrt{3}-4}{2}$$

Divide -4+10\sqrt{3} by 2.

$$x=5\sqrt{3}-2$$

Now solve the equation x=\frac{-4±10\sqrt{3}}{2} when ± is minus. Subtract 10\sqrt{3} from -4.

$$x=\frac{-10\sqrt{3}-4}{2}$$

Divide -4-10\sqrt{3} by 2.

$$x=-5\sqrt{3}-2$$

The equation is now solved.

$$x=5\sqrt{3}-2 x=-5\sqrt{3}-2$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$75=x^{2}+4x+4$$

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

$$x^{2}+4x+4=75$$

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}.

$$\left(x+2\right)^{2}=75$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{75}$$

Simplify.

$$x+2=5\sqrt{3} x+2=-5\sqrt{3}$$

Subtract 2 from both sides of the equation.

$$x=5\sqrt{3}-2 x=-5\sqrt{3}-2$$