Solve 25(x+2)^2=(x-7)^2-81

$$25 ( x + 2 ) ^ { 2 } = ( x - 7 ) ^ { 2 } - 81$$

Solve for x

$$x = -\frac{11}{4} = -2\frac{3}{4} = -2.75$$

$$x=-2$$

Steps Using the Quadratic Formula

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

$$25\left(x^{2}+4x+4\right)=\left(x-7\right)^{2}-81$$

Use the distributive property to multiply 25 by x^{2}+4x+4.

$$25x^{2}+100x+100=\left(x-7\right)^{2}-81$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-7\right)^{2}.

$$25x^{2}+100x+100=x^{2}-14x+49-81$$

Subtract 81 from 49 to get -32.

$$25x^{2}+100x+100=x^{2}-14x-32$$

Subtract x^{2} from both sides.

$$25x^{2}+100x+100-x^{2}=-14x-32$$

Combine 25x^{2} and -x^{2} to get 24x^{2}.

$$24x^{2}+100x+100=-14x-32$$

Add 14x to both sides.

$$24x^{2}+100x+100+14x=-32$$

Combine 100x and 14x to get 114x.

$$24x^{2}+114x+100=-32$$

Add 32 to both sides.

$$24x^{2}+114x+100+32=0$$

Add 100 and 32 to get 132.

$$24x^{2}+114x+132=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 114 for b, and 132 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-114±\sqrt{114^{2}-4\times 24\times 132}}{2\times 24}$$

Square 114.

$$x=\frac{-114±\sqrt{12996-4\times 24\times 132}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-114±\sqrt{12996-96\times 132}}{2\times 24}$$

Multiply -96 times 132.

$$x=\frac{-114±\sqrt{12996-12672}}{2\times 24}$$

Add 12996 to -12672.

$$x=\frac{-114±\sqrt{324}}{2\times 24}$$

Take the square root of 324.

$$x=\frac{-114±18}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{-114±18}{48}$$

Now solve the equation x=\frac{-114±18}{48} when ± is plus. Add -114 to 18.

$$x=-\frac{96}{48}$$

Divide -96 by 48.

$$x=-2$$

Now solve the equation x=\frac{-114±18}{48} when ± is minus. Subtract 18 from -114.

$$x=-\frac{132}{48}$$

Reduce the fraction \frac{-132}{48} to lowest terms by extracting and canceling out 12.

$$x=-\frac{11}{4}$$

The equation is now solved.

$$x=-2 x=-\frac{11}{4}$$

Steps for Completing the Square

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

$$25\left(x^{2}+4x+4\right)=\left(x-7\right)^{2}-81$$

Use the distributive property to multiply 25 by x^{2}+4x+4.

$$25x^{2}+100x+100=\left(x-7\right)^{2}-81$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-7\right)^{2}.

$$25x^{2}+100x+100=x^{2}-14x+49-81$$

Subtract 81 from 49 to get -32.

$$25x^{2}+100x+100=x^{2}-14x-32$$

Subtract x^{2} from both sides.

$$25x^{2}+100x+100-x^{2}=-14x-32$$

Combine 25x^{2} and -x^{2} to get 24x^{2}.

$$24x^{2}+100x+100=-14x-32$$

Add 14x to both sides.

$$24x^{2}+100x+100+14x=-32$$

Combine 100x and 14x to get 114x.

$$24x^{2}+114x+100=-32$$

Subtract 100 from both sides.

$$24x^{2}+114x=-32-100$$

Subtract 100 from -32 to get -132.

$$24x^{2}+114x=-132$$

Divide both sides by 24.

$$\frac{24x^{2}+114x}{24}=-\frac{132}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\frac{114}{24}x=-\frac{132}{24}$$

Reduce the fraction \frac{114}{24} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{19}{4}x=-\frac{132}{24}$$

Reduce the fraction \frac{-132}{24} to lowest terms by extracting and canceling out 12.

$$x^{2}+\frac{19}{4}x=-\frac{11}{2}$$

Divide \frac{19}{4}, the coefficient of the x term, by 2 to get \frac{19}{8}. Then add the square of \frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{19}{4}x+\left(\frac{19}{8}\right)^{2}=-\frac{11}{2}+\left(\frac{19}{8}\right)^{2}$$

Square \frac{19}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=-\frac{11}{2}+\frac{361}{64}$$

Add -\frac{11}{2} to \frac{361}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=\frac{9}{64}$$

Factor x^{2}+\frac{19}{4}x+\frac{361}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{19}{8}\right)^{2}=\frac{9}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{19}{8}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Simplify.

$$x+\frac{19}{8}=\frac{3}{8} x+\frac{19}{8}=-\frac{3}{8}$$

Subtract \frac{19}{8} from both sides of the equation.

$$x=-2 x=-\frac{11}{4}$$