Solve E=(4x-1)^2-(3x+2)(4x-1)

$$E = ( 4 x - 1 ) ^ { 2 } - ( 3 x + 2 ) ( 4 x - 1 )$$

Solve for E

$$E=\left(x-3\right)\left(4x-1\right)$$

Solution Steps

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

$$E=16x^{2}-8x+1-\left(3x+2\right)\left(4x-1\right)$$

Use the distributive property to multiply 3x+2 by 4x-1 and combine like terms.

$$E=16x^{2}-8x+1-\left(12x^{2}+5x-2\right)$$

To find the opposite of 12x^{2}+5x-2, find the opposite of each term.

$$E=16x^{2}-8x+1-12x^{2}-5x+2$$

Combine 16x^{2} and -12x^{2} to get 4x^{2}.

$$E=4x^{2}-8x+1-5x+2$$

Combine -8x and -5x to get -13x.

$$E=4x^{2}-13x+1+2$$

Add 1 and 2 to get 3.

$$E=4x^{2}-13x+3$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{16E+121}+13}{8}$$

$$x=\frac{\sqrt{16E+121}+13}{8}$$

Solve for x

$$x=\frac{-\sqrt{16E+121}+13}{8}$$

$$x=\frac{\sqrt{16E+121}+13}{8}\text{, }E\geq -\frac{121}{16}$$