Solve (x-1)^2+11x+199=3x^2-(x-2)^2

$$( x - 1 ) ^ { 2 } + 11 x + 199 = 3 x ^ { 2 } - ( x - 2 ) ^ { 2 }$$

Solve for x

$$x=-12$$

$$x=17$$

Steps Using Factoring By Grouping

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

$$x^{2}-2x+1+11x+199=3x^{2}-\left(x-2\right)^{2}$$

Combine -2x and 11x to get 9x.

$$x^{2}+9x+1+199=3x^{2}-\left(x-2\right)^{2}$$

Add 1 and 199 to get 200.

$$x^{2}+9x+200=3x^{2}-\left(x-2\right)^{2}$$

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

$$x^{2}+9x+200=3x^{2}-\left(x^{2}-4x+4\right)$$

To find the opposite of x^{2}-4x+4, find the opposite of each term.

$$x^{2}+9x+200=3x^{2}-x^{2}+4x-4$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$x^{2}+9x+200=2x^{2}+4x-4$$

Subtract 2x^{2} from both sides.

$$x^{2}+9x+200-2x^{2}=4x-4$$

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

$$-x^{2}+9x+200=4x-4$$

Subtract 4x from both sides.

$$-x^{2}+9x+200-4x=-4$$

Combine 9x and -4x to get 5x.

$$-x^{2}+5x+200=-4$$

Add 4 to both sides.

$$-x^{2}+5x+200+4=0$$

Add 200 and 4 to get 204.

$$-x^{2}+5x+204=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+204. To find a and b, set up a system to be solved.

$$a+b=5 ab=-204=-204$$

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 -204.

$$-1,204 -2,102 -3,68 -4,51 -6,34 -12,17$$

Calculate the sum for each pair.

$$-1+204=203 -2+102=100 -3+68=65 -4+51=47 -6+34=28 -12+17=5$$

The solution is the pair that gives sum 5.

$$a=17 b=-12$$

Rewrite -x^{2}+5x+204 as \left(-x^{2}+17x\right)+\left(-12x+204\right).

$$\left(-x^{2}+17x\right)+\left(-12x+204\right)$$

Factor out -x in the first and -12 in the second group.

$$-x\left(x-17\right)-12\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(-x-12\right)$$

To find equation solutions, solve x-17=0 and -x-12=0.

$$x=17 x=-12$$

Steps Using the Quadratic Formula

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

$$x^{2}-2x+1+11x+199=3x^{2}-\left(x-2\right)^{2}$$

Combine -2x and 11x to get 9x.

$$x^{2}+9x+1+199=3x^{2}-\left(x-2\right)^{2}$$

Add 1 and 199 to get 200.

$$x^{2}+9x+200=3x^{2}-\left(x-2\right)^{2}$$

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

$$x^{2}+9x+200=3x^{2}-\left(x^{2}-4x+4\right)$$

To find the opposite of x^{2}-4x+4, find the opposite of each term.

$$x^{2}+9x+200=3x^{2}-x^{2}+4x-4$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$x^{2}+9x+200=2x^{2}+4x-4$$

Subtract 2x^{2} from both sides.

$$x^{2}+9x+200-2x^{2}=4x-4$$

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

$$-x^{2}+9x+200=4x-4$$

Subtract 4x from both sides.

$$-x^{2}+9x+200-4x=-4$$

Combine 9x and -4x to get 5x.

$$-x^{2}+5x+200=-4$$

Add 4 to both sides.

$$-x^{2}+5x+200+4=0$$

Add 200 and 4 to get 204.

$$-x^{2}+5x+204=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and 204 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 204}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\times 204}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\times 204}}{2\left(-1\right)}$$

Multiply 4 times 204.

$$x=\frac{-5±\sqrt{25+816}}{2\left(-1\right)}$$

Add 25 to 816.

$$x=\frac{-5±\sqrt{841}}{2\left(-1\right)}$$

Take the square root of 841.

$$x=\frac{-5±29}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±29}{-2}$$

Now solve the equation x=\frac{-5±29}{-2} when ± is plus. Add -5 to 29.

$$x=\frac{24}{-2}$$

Divide 24 by -2.

$$x=-12$$

Now solve the equation x=\frac{-5±29}{-2} when ± is minus. Subtract 29 from -5.

$$x=-\frac{34}{-2}$$

Divide -34 by -2.

$$x=17$$

The equation is now solved.

$$x=-12 x=17$$

Steps for Completing the Square

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

$$x^{2}-2x+1+11x+199=3x^{2}-\left(x-2\right)^{2}$$

Combine -2x and 11x to get 9x.

$$x^{2}+9x+1+199=3x^{2}-\left(x-2\right)^{2}$$

Add 1 and 199 to get 200.

$$x^{2}+9x+200=3x^{2}-\left(x-2\right)^{2}$$

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

$$x^{2}+9x+200=3x^{2}-\left(x^{2}-4x+4\right)$$

To find the opposite of x^{2}-4x+4, find the opposite of each term.

$$x^{2}+9x+200=3x^{2}-x^{2}+4x-4$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$x^{2}+9x+200=2x^{2}+4x-4$$

Subtract 2x^{2} from both sides.

$$x^{2}+9x+200-2x^{2}=4x-4$$

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

$$-x^{2}+9x+200=4x-4$$

Subtract 4x from both sides.

$$-x^{2}+9x+200-4x=-4$$

Combine 9x and -4x to get 5x.

$$-x^{2}+5x+200=-4$$

Subtract 200 from both sides.

$$-x^{2}+5x=-4-200$$

Subtract 200 from -4 to get -204.

$$-x^{2}+5x=-204$$

Divide both sides by -1.

$$\frac{-x^{2}+5x}{-1}=-\frac{204}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{5}{-1}x=-\frac{204}{-1}$$

Divide 5 by -1.

$$x^{2}-5x=-\frac{204}{-1}$$

Divide -204 by -1.

$$x^{2}-5x=204$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=204+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=204+\frac{25}{4}$$

Add 204 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{841}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{841}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{841}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{29}{2} x-\frac{5}{2}=-\frac{29}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=17 x=-12$$