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

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

Solve for r

$$r=\left(x-3\right)\left(5x-1\right)$$

Solution Steps

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

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

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

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

To find the opposite of 4x^{2}+4x+1, find the opposite of each term.

$$r=9x^{2}-12x+4-4x^{2}-4x-1$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$r=5x^{2}-12x+4-4x-1$$

Combine -12x and -4x to get -16x.

$$r=5x^{2}-16x+4-1$$

Subtract 1 from 4 to get 3.

$$r=5x^{2}-16x+3$$

Solve for x

$$x=\frac{-\sqrt{5r+49}+8}{5}$$

$$x=\frac{\sqrt{5r+49}+8}{5}\text{, }r\geq -\frac{49}{5}$$