Solve 9(2m+3)^2=(2m-5)^2

$$9 ( 2 m + 3 ) ^ { 2 } = ( 2 m - 5 ) ^ { 2 }$$

Solve for m

$$m = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$m=-\frac{1}{2}=-0.5$$

Steps Using the Quadratic Formula

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

$$9\left(4m^{2}+12m+9\right)=\left(2m-5\right)^{2}$$

Use the distributive property to multiply 9 by 4m^{2}+12m+9.

$$36m^{2}+108m+81=\left(2m-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2m-5\right)^{2}.

$$36m^{2}+108m+81=4m^{2}-20m+25$$

Subtract 4m^{2} from both sides.

$$36m^{2}+108m+81-4m^{2}=-20m+25$$

Combine 36m^{2} and -4m^{2} to get 32m^{2}.

$$32m^{2}+108m+81=-20m+25$$

Add 20m to both sides.

$$32m^{2}+108m+81+20m=25$$

Combine 108m and 20m to get 128m.

$$32m^{2}+128m+81=25$$

Subtract 25 from both sides.

$$32m^{2}+128m+81-25=0$$

Subtract 25 from 81 to get 56.

$$32m^{2}+128m+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 32 for a, 128 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-128±\sqrt{128^{2}-4\times 32\times 56}}{2\times 32}$$

Square 128.

$$m=\frac{-128±\sqrt{16384-4\times 32\times 56}}{2\times 32}$$

Multiply -4 times 32.

$$m=\frac{-128±\sqrt{16384-128\times 56}}{2\times 32}$$

Multiply -128 times 56.

$$m=\frac{-128±\sqrt{16384-7168}}{2\times 32}$$

Add 16384 to -7168.

$$m=\frac{-128±\sqrt{9216}}{2\times 32}$$

Take the square root of 9216.

$$m=\frac{-128±96}{2\times 32}$$

Multiply 2 times 32.

$$m=\frac{-128±96}{64}$$

Now solve the equation m=\frac{-128±96}{64} when ± is plus. Add -128 to 96.

$$m=-\frac{32}{64}$$

Reduce the fraction \frac{-32}{64} to lowest terms by extracting and canceling out 32.

$$m=-\frac{1}{2}$$

Now solve the equation m=\frac{-128±96}{64} when ± is minus. Subtract 96 from -128.

$$m=-\frac{224}{64}$$

Reduce the fraction \frac{-224}{64} to lowest terms by extracting and canceling out 32.

$$m=-\frac{7}{2}$$

The equation is now solved.

$$m=-\frac{1}{2} m=-\frac{7}{2}$$

Steps for Completing the Square

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

$$9\left(4m^{2}+12m+9\right)=\left(2m-5\right)^{2}$$

Use the distributive property to multiply 9 by 4m^{2}+12m+9.

$$36m^{2}+108m+81=\left(2m-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2m-5\right)^{2}.

$$36m^{2}+108m+81=4m^{2}-20m+25$$

Subtract 4m^{2} from both sides.

$$36m^{2}+108m+81-4m^{2}=-20m+25$$

Combine 36m^{2} and -4m^{2} to get 32m^{2}.

$$32m^{2}+108m+81=-20m+25$$

Add 20m to both sides.

$$32m^{2}+108m+81+20m=25$$

Combine 108m and 20m to get 128m.

$$32m^{2}+128m+81=25$$

Subtract 81 from both sides.

$$32m^{2}+128m=25-81$$

Subtract 81 from 25 to get -56.

$$32m^{2}+128m=-56$$

Divide both sides by 32.

$$\frac{32m^{2}+128m}{32}=-\frac{56}{32}$$

Dividing by 32 undoes the multiplication by 32.

$$m^{2}+\frac{128}{32}m=-\frac{56}{32}$$

Divide 128 by 32.

$$m^{2}+4m=-\frac{56}{32}$$

Reduce the fraction \frac{-56}{32} to lowest terms by extracting and canceling out 8.

$$m^{2}+4m=-\frac{7}{4}$$

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

$$m^{2}+4m+2^{2}=-\frac{7}{4}+2^{2}$$

Square 2.

$$m^{2}+4m+4=-\frac{7}{4}+4$$

Add -\frac{7}{4} to 4.

$$m^{2}+4m+4=\frac{9}{4}$$

Factor m^{2}+4m+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}.

$$\left(m+2\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+2\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$m+2=\frac{3}{2} m+2=-\frac{3}{2}$$

Subtract 2 from both sides of the equation.

$$m=-\frac{1}{2} m=-\frac{7}{2}$$