Solve 50left(1+xright)+left(1+xright)^2=450

$$50 \left( 1+x \right) + { \left(1+x \right) }^{ 2 } =450$$

Solve for x

$$x=5\sqrt{43}-26\approx 6.787192622$$

$$x=-5\sqrt{43}-26\approx -58.787192622$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 50 by 1+x.

$$50+50x+\left(1+x\right)^{2}=450$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$50+50x+1+2x+x^{2}=450$$

Add 50 and 1 to get 51.

$$51+50x+2x+x^{2}=450$$

Combine 50x and 2x to get 52x.

$$51+52x+x^{2}=450$$

Subtract 450 from both sides.

$$51+52x+x^{2}-450=0$$

Subtract 450 from 51 to get -399.

$$-399+52x+x^{2}=0$$

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.

$$x^{2}+52x-399=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 52 for b, and -399 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-52±\sqrt{52^{2}-4\left(-399\right)}}{2}$$

Square 52.

$$x=\frac{-52±\sqrt{2704-4\left(-399\right)}}{2}$$

Multiply -4 times -399.

$$x=\frac{-52±\sqrt{2704+1596}}{2}$$

Add 2704 to 1596.

$$x=\frac{-52±\sqrt{4300}}{2}$$

Take the square root of 4300.

$$x=\frac{-52±10\sqrt{43}}{2}$$

Now solve the equation x=\frac{-52±10\sqrt{43}}{2} when ± is plus. Add -52 to 10\sqrt{43}.

$$x=\frac{10\sqrt{43}-52}{2}$$

Divide -52+10\sqrt{43} by 2.

$$x=5\sqrt{43}-26$$

Now solve the equation x=\frac{-52±10\sqrt{43}}{2} when ± is minus. Subtract 10\sqrt{43} from -52.

$$x=\frac{-10\sqrt{43}-52}{2}$$

Divide -52-10\sqrt{43} by 2.

$$x=-5\sqrt{43}-26$$

The equation is now solved.

$$x=5\sqrt{43}-26 x=-5\sqrt{43}-26$$

Steps for Completing the Square

Use the distributive property to multiply 50 by 1+x.

$$50+50x+\left(1+x\right)^{2}=450$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$50+50x+1+2x+x^{2}=450$$

Add 50 and 1 to get 51.

$$51+50x+2x+x^{2}=450$$

Combine 50x and 2x to get 52x.

$$51+52x+x^{2}=450$$

Subtract 51 from both sides.

$$52x+x^{2}=450-51$$

Subtract 51 from 450 to get 399.

$$52x+x^{2}=399$$

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.

$$x^{2}+52x=399$$

Divide 52, the coefficient of the x term, by 2 to get 26. Then add the square of 26 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+52x+26^{2}=399+26^{2}$$

Square 26.

$$x^{2}+52x+676=399+676$$

Add 399 to 676.

$$x^{2}+52x+676=1075$$

Factor x^{2}+52x+676. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+26\right)^{2}=1075$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+26\right)^{2}}=\sqrt{1075}$$

Simplify.

$$x+26=5\sqrt{43} x+26=-5\sqrt{43}$$

Subtract 26 from both sides of the equation.

$$x=5\sqrt{43}-26 x=-5\sqrt{43}-26$$