Solve M=(2x-1)^2-(x+3)^2

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

Solve for M

$$M=\left(x-4\right)\left(3x+2\right)$$

Solution Steps

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

$$M=4x^{2}-4x+1-\left(x+3\right)^{2}$$

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

$$M=4x^{2}-4x+1-\left(x^{2}+6x+9\right)$$

To find the opposite of x^{2}+6x+9, find the opposite of each term.

$$M=4x^{2}-4x+1-x^{2}-6x-9$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$M=3x^{2}-4x+1-6x-9$$

Combine -4x and -6x to get -10x.

$$M=3x^{2}-10x+1-9$$

Subtract 9 from 1 to get -8.

$$M=3x^{2}-10x-8$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{3M+49}+5}{3}$$

$$x=\frac{\sqrt{3M+49}+5}{3}$$

Solve for x

$$x=\frac{-\sqrt{3M+49}+5}{3}$$

$$x=\frac{\sqrt{3M+49}+5}{3}\text{, }M\geq -\frac{49}{3}$$