Solve D=(4x-5)(4x+5)-(27-20x+9x^2)

$$D = ( 4 x - 5 ) ( 4 x + 5 ) - ( 27 - 20 x + 9 x ^ { 2 } )$$

Solve for D

$$D=7x^{2}+20x-52$$

Solution Steps

Consider \left(4x-5\right)\left(4x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$D=\left(4x\right)^{2}-25-\left(27-20x+9x^{2}\right)$$

Expand \left(4x\right)^{2}.

$$D=4^{2}x^{2}-25-\left(27-20x+9x^{2}\right)$$

Calculate 4 to the power of 2 and get 16.

$$D=16x^{2}-25-\left(27-20x+9x^{2}\right)$$

To find the opposite of 27-20x+9x^{2}, find the opposite of each term.

$$D=16x^{2}-25-27+20x-9x^{2}$$

Subtract 27 from -25 to get -52.

$$D=16x^{2}-52+20x-9x^{2}$$

Combine 16x^{2} and -9x^{2} to get 7x^{2}.

$$D=7x^{2}-52+20x$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{7D+464}-10}{7}$$

$$x=\frac{\sqrt{7D+464}-10}{7}$$

Solve for x

$$x=\frac{-\sqrt{7D+464}-10}{7}$$

$$x=\frac{\sqrt{7D+464}-10}{7}\text{, }D\geq -\frac{464}{7}$$