Solve (50+x)^2=11-3

$$( 50 + x ) ^ { 2 } = 11 - 3$$

Solve for x

$$x=2\sqrt{2}-50\approx -47.171572875$$

$$x=-2\sqrt{2}-50\approx -52.828427125$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(50+x\right)^{2}.

$$2500+100x+x^{2}=11-3$$

Subtract 3 from 11 to get 8.

$$2500+100x+x^{2}=8$$

Subtract 8 from both sides.

$$2500+100x+x^{2}-8=0$$

Subtract 8 from 2500 to get 2492.

$$2492+100x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+100x+2492=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 100 for b, and 2492 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\times 2492}}{2}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\times 2492}}{2}$$

Multiply -4 times 2492.

$$x=\frac{-100±\sqrt{10000-9968}}{2}$$

Add 10000 to -9968.

$$x=\frac{-100±\sqrt{32}}{2}$$

Take the square root of 32.

$$x=\frac{-100±4\sqrt{2}}{2}$$

Now solve the equation x=\frac{-100±4\sqrt{2}}{2} when ± is plus. Add -100 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}-100}{2}$$

Divide -100+4\sqrt{2} by 2.

$$x=2\sqrt{2}-50$$

Now solve the equation x=\frac{-100±4\sqrt{2}}{2} when ± is minus. Subtract 4\sqrt{2} from -100.

$$x=\frac{-4\sqrt{2}-100}{2}$$

Divide -100-4\sqrt{2} by 2.

$$x=-2\sqrt{2}-50$$

The equation is now solved.

$$x=2\sqrt{2}-50 x=-2\sqrt{2}-50$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(50+x\right)^{2}.

$$2500+100x+x^{2}=11-3$$

Subtract 3 from 11 to get 8.

$$2500+100x+x^{2}=8$$

Subtract 2500 from both sides.

$$100x+x^{2}=8-2500$$

Subtract 2500 from 8 to get -2492.

$$100x+x^{2}=-2492$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+100x=-2492$$

Divide 100, the coefficient of the x term, by 2 to get 50. Then add the square of 50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+100x+50^{2}=-2492+50^{2}$$

Square 50.

$$x^{2}+100x+2500=-2492+2500$$

Add -2492 to 2500.

$$x^{2}+100x+2500=8$$

Factor x^{2}+100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+50\right)^{2}=8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{8}$$

Simplify.

$$x+50=2\sqrt{2} x+50=-2\sqrt{2}$$

Subtract 50 from both sides of the equation.

$$x=2\sqrt{2}-50 x=-2\sqrt{2}-50$$