Solve A=(4x-1)^2-(x-5)^2

$$A = ( 4 x - 1 ) ^ { 2 } - ( x - 5 ) ^ { 2 }$$

Solve for A

$$A=\left(5x-6\right)\left(3x+4\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-1\right)^{2}.

$$A=16x^{2}-8x+1-\left(x-5\right)^{2}$$

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

$$A=16x^{2}-8x+1-\left(x^{2}-10x+25\right)$$

To find the opposite of x^{2}-10x+25, find the opposite of each term.

$$A=16x^{2}-8x+1-x^{2}+10x-25$$

Combine 16x^{2} and -x^{2} to get 15x^{2}.

$$A=15x^{2}-8x+1+10x-25$$

Combine -8x and 10x to get 2x.

$$A=15x^{2}+2x+1-25$$

Subtract 25 from 1 to get -24.

$$A=15x^{2}+2x-24$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{15A+361}-1}{15}$$

$$x=\frac{\sqrt{15A+361}-1}{15}$$

Solve for x

$$x=\frac{-\sqrt{15A+361}-1}{15}$$

$$x=\frac{\sqrt{15A+361}-1}{15}\text{, }A\geq -\frac{361}{15}$$