Solve (2x+3)^2-15^2=10^2-(x-1)^2

$$( 2 x + 3 ) ^ { 2 } - 15 ^ { 2 } = 10 ^ { 2 } - ( x - 1 ) ^ { 2 }$$

Solve for x

$$x=-9$$

$$x=7$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$4x^{2}+12x+9-15^{2}=10^{2}-\left(x-1\right)^{2}$$

Calculate 15 to the power of 2 and get 225.

$$4x^{2}+12x+9-225=10^{2}-\left(x-1\right)^{2}$$

Subtract 225 from 9 to get -216.

$$4x^{2}+12x-216=10^{2}-\left(x-1\right)^{2}$$

Calculate 10 to the power of 2 and get 100.

$$4x^{2}+12x-216=100-\left(x-1\right)^{2}$$

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

$$4x^{2}+12x-216=100-\left(x^{2}-2x+1\right)$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$4x^{2}+12x-216=100-x^{2}+2x-1$$

Subtract 1 from 100 to get 99.

$$4x^{2}+12x-216=99-x^{2}+2x$$

Subtract 99 from both sides.

$$4x^{2}+12x-216-99=-x^{2}+2x$$

Subtract 99 from -216 to get -315.

$$4x^{2}+12x-315=-x^{2}+2x$$

Add x^{2} to both sides.

$$4x^{2}+12x-315+x^{2}=2x$$

Combine 4x^{2} and x^{2} to get 5x^{2}.

$$5x^{2}+12x-315=2x$$

Subtract 2x from both sides.

$$5x^{2}+12x-315-2x=0$$

Combine 12x and -2x to get 10x.

$$5x^{2}+10x-315=0$$

Divide both sides by 5.

$$x^{2}+2x-63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-63. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-63\right)=-63$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -63.

$$-1,63 -3,21 -7,9$$

Calculate the sum for each pair.

$$-1+63=62 -3+21=18 -7+9=2$$

The solution is the pair that gives sum 2.

$$a=-7 b=9$$

Rewrite x^{2}+2x-63 as \left(x^{2}-7x\right)+\left(9x-63\right).

$$\left(x^{2}-7x\right)+\left(9x-63\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-7\right)+9\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(x+9\right)$$

To find equation solutions, solve x-7=0 and x+9=0.

$$x=7 x=-9$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$4x^{2}+12x+9-15^{2}=10^{2}-\left(x-1\right)^{2}$$

Calculate 15 to the power of 2 and get 225.

$$4x^{2}+12x+9-225=10^{2}-\left(x-1\right)^{2}$$

Subtract 225 from 9 to get -216.

$$4x^{2}+12x-216=10^{2}-\left(x-1\right)^{2}$$

Calculate 10 to the power of 2 and get 100.

$$4x^{2}+12x-216=100-\left(x-1\right)^{2}$$

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

$$4x^{2}+12x-216=100-\left(x^{2}-2x+1\right)$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$4x^{2}+12x-216=100-x^{2}+2x-1$$

Subtract 1 from 100 to get 99.

$$4x^{2}+12x-216=99-x^{2}+2x$$

Subtract 99 from both sides.

$$4x^{2}+12x-216-99=-x^{2}+2x$$

Subtract 99 from -216 to get -315.

$$4x^{2}+12x-315=-x^{2}+2x$$

Add x^{2} to both sides.

$$4x^{2}+12x-315+x^{2}=2x$$

Combine 4x^{2} and x^{2} to get 5x^{2}.

$$5x^{2}+12x-315=2x$$

Subtract 2x from both sides.

$$5x^{2}+12x-315-2x=0$$

Combine 12x and -2x to get 10x.

$$5x^{2}+10x-315=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 10 for b, and -315 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-315\right)}}{2\times 5}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-315\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-10±\sqrt{100-20\left(-315\right)}}{2\times 5}$$

Multiply -20 times -315.

$$x=\frac{-10±\sqrt{100+6300}}{2\times 5}$$

Add 100 to 6300.

$$x=\frac{-10±\sqrt{6400}}{2\times 5}$$

Take the square root of 6400.

$$x=\frac{-10±80}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-10±80}{10}$$

Now solve the equation x=\frac{-10±80}{10} when ± is plus. Add -10 to 80.

$$x=\frac{70}{10}$$

Divide 70 by 10.

$$x=7$$

Now solve the equation x=\frac{-10±80}{10} when ± is minus. Subtract 80 from -10.

$$x=-\frac{90}{10}$$

Divide -90 by 10.

$$x=-9$$

The equation is now solved.

$$x=7 x=-9$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$4x^{2}+12x+9-15^{2}=10^{2}-\left(x-1\right)^{2}$$

Calculate 15 to the power of 2 and get 225.

$$4x^{2}+12x+9-225=10^{2}-\left(x-1\right)^{2}$$

Subtract 225 from 9 to get -216.

$$4x^{2}+12x-216=10^{2}-\left(x-1\right)^{2}$$

Calculate 10 to the power of 2 and get 100.

$$4x^{2}+12x-216=100-\left(x-1\right)^{2}$$

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

$$4x^{2}+12x-216=100-\left(x^{2}-2x+1\right)$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$4x^{2}+12x-216=100-x^{2}+2x-1$$

Subtract 1 from 100 to get 99.

$$4x^{2}+12x-216=99-x^{2}+2x$$

Add x^{2} to both sides.

$$4x^{2}+12x-216+x^{2}=99+2x$$

Combine 4x^{2} and x^{2} to get 5x^{2}.

$$5x^{2}+12x-216=99+2x$$

Subtract 2x from both sides.

$$5x^{2}+12x-216-2x=99$$

Combine 12x and -2x to get 10x.

$$5x^{2}+10x-216=99$$

Add 216 to both sides.

$$5x^{2}+10x=99+216$$

Add 99 and 216 to get 315.

$$5x^{2}+10x=315$$

Divide both sides by 5.

$$\frac{5x^{2}+10x}{5}=\frac{315}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{10}{5}x=\frac{315}{5}$$

Divide 10 by 5.

$$x^{2}+2x=\frac{315}{5}$$

Divide 315 by 5.

$$x^{2}+2x=63$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=63+1^{2}$$

Square 1.

$$x^{2}+2x+1=63+1$$

Add 63 to 1.

$$x^{2}+2x+1=64$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{64}$$

Simplify.

$$x+1=8 x+1=-8$$

Subtract 1 from both sides of the equation.

$$x=7 x=-9$$