Solve L=(2x-3)^2-5(2x-3)(x+1)

$$L = ( 2 x - 3 ) ^ { 2 } - 5 ( 2 x - 3 ) ( x + 1 )$$

Solve for L

$$L=24-7x-6x^{2}$$

Solution Steps

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

$$L=4x^{2}-12x+9-5\left(2x-3\right)\left(x+1\right)$$

Use the distributive property to multiply -5 by 2x-3.

$$L=4x^{2}-12x+9+\left(-10x+15\right)\left(x+1\right)$$

Use the distributive property to multiply -10x+15 by x+1 and combine like terms.

$$L=4x^{2}-12x+9-10x^{2}+5x+15$$

Combine 4x^{2} and -10x^{2} to get -6x^{2}.

$$L=-6x^{2}-12x+9+5x+15$$

Combine -12x and 5x to get -7x.

$$L=-6x^{2}-7x+9+15$$

Add 9 and 15 to get 24.

$$L=-6x^{2}-7x+24$$

Solve for x (complex solution)

$$x=\frac{\sqrt{625-24L}-7}{12}$$

$$x=\frac{-\sqrt{625-24L}-7}{12}$$

Solve for x

$$x=\frac{\sqrt{625-24L}-7}{12}$$

$$x=\frac{-\sqrt{625-24L}-7}{12}\text{, }L\leq \frac{625}{24}$$