Solve x^2+18x+35=-x^2+2x+5

$${ x }^{ 2 } +18x+35=- { x }^{ 2 } +2x+5$$

Solve for x

$$x=-5$$

$$x=-3$$

Steps Using Factoring By Grouping

Add x^{2} to both sides.

$$x^{2}+18x+35+x^{2}=2x+5$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+18x+35=2x+5$$

Subtract 2x from both sides.

$$2x^{2}+18x+35-2x=5$$

Combine 18x and -2x to get 16x.

$$2x^{2}+16x+35=5$$

Subtract 5 from both sides.

$$2x^{2}+16x+35-5=0$$

Subtract 5 from 35 to get 30.

$$2x^{2}+16x+30=0$$

Divide both sides by 2.

$$x^{2}+8x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 15=15$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

Rewrite x^{2}+8x+15 as \left(x^{2}+3x\right)+\left(5x+15\right).

$$\left(x^{2}+3x\right)+\left(5x+15\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x+3\right)+5\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+5\right)$$

To find equation solutions, solve x+3=0 and x+5=0.

$$x=-3 x=-5$$

Steps Using the Quadratic Formula

Add x^{2} to both sides.

$$x^{2}+18x+35+x^{2}=2x+5$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+18x+35=2x+5$$

Subtract 2x from both sides.

$$2x^{2}+18x+35-2x=5$$

Combine 18x and -2x to get 16x.

$$2x^{2}+16x+35=5$$

Subtract 5 from both sides.

$$2x^{2}+16x+35-5=0$$

Subtract 5 from 35 to get 30.

$$2x^{2}+16x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 16 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\times 30}}{2\times 2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 2\times 30}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-16±\sqrt{256-8\times 30}}{2\times 2}$$

Multiply -8 times 30.

$$x=\frac{-16±\sqrt{256-240}}{2\times 2}$$

Add 256 to -240.

$$x=\frac{-16±\sqrt{16}}{2\times 2}$$

Take the square root of 16.

$$x=\frac{-16±4}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-16±4}{4}$$

Now solve the equation x=\frac{-16±4}{4} when ± is plus. Add -16 to 4.

$$x=-\frac{12}{4}$$

Divide -12 by 4.

$$x=-3$$

Now solve the equation x=\frac{-16±4}{4} when ± is minus. Subtract 4 from -16.

$$x=-\frac{20}{4}$$

Divide -20 by 4.

$$x=-5$$

The equation is now solved.

$$x=-3 x=-5$$

Steps for Completing the Square

Add x^{2} to both sides.

$$x^{2}+18x+35+x^{2}=2x+5$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+18x+35=2x+5$$

Subtract 2x from both sides.

$$2x^{2}+18x+35-2x=5$$

Combine 18x and -2x to get 16x.

$$2x^{2}+16x+35=5$$

Subtract 35 from both sides.

$$2x^{2}+16x=5-35$$

Subtract 35 from 5 to get -30.

$$2x^{2}+16x=-30$$

Divide both sides by 2.

$$\frac{2x^{2}+16x}{2}=-\frac{30}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{16}{2}x=-\frac{30}{2}$$

Divide 16 by 2.

$$x^{2}+8x=-\frac{30}{2}$$

Divide -30 by 2.

$$x^{2}+8x=-15$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-15+4^{2}$$

Square 4.

$$x^{2}+8x+16=-15+16$$

Add -15 to 16.

$$x^{2}+8x+16=1$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{1}$$

Simplify.

$$x+4=1 x+4=-1$$

Subtract 4 from both sides of the equation.

$$x=-3 x=-5$$