Solve E=(5x-2)^2-(7x-4)^2

$$E = ( 5 x - 2 ) ^ { 2 } - ( 7 x - 4 ) ^ { 2 }$$

Solve for E

$$E=-12\left(x-1\right)\left(2x-1\right)$$

Solution Steps

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

$$E=25x^{2}-20x+4-\left(7x-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7x-4\right)^{2}.

$$E=25x^{2}-20x+4-\left(49x^{2}-56x+16\right)$$

To find the opposite of 49x^{2}-56x+16, find the opposite of each term.

$$E=25x^{2}-20x+4-49x^{2}+56x-16$$

Combine 25x^{2} and -49x^{2} to get -24x^{2}.

$$E=-24x^{2}-20x+4+56x-16$$

Combine -20x and 56x to get 36x.

$$E=-24x^{2}+36x+4-16$$

Subtract 16 from 4 to get -12.

$$E=-24x^{2}+36x-12$$

Solve for x (complex solution)

$$x=\frac{\sqrt{9-6E}}{12}+\frac{3}{4}$$

$$x=-\frac{\sqrt{9-6E}}{12}+\frac{3}{4}$$

Solve for x

$$x=\frac{\sqrt{9-6E}}{12}+\frac{3}{4}$$

$$x=-\frac{\sqrt{9-6E}}{12}+\frac{3}{4}\text{, }E\leq \frac{3}{2}$$