Solve 4x^2+20x+25-x^2-8x+3x-24

$$4 { x }^{ 2 } +20x+25- { x }^{ 2 } -8x+3x-24$$

Evaluate

$$3x^{2}+15x+1$$

Solution Steps

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+20x+25-8x+3x-24$$

Combine 20x and -8x to get 12x.

$$3x^{2}+12x+25+3x-24$$

Combine 12x and 3x to get 15x.

$$3x^{2}+15x+25-24$$

Subtract 24 from 25 to get 1.

$$3x^{2}+15x+1$$

Factor

$$3\left(x-\left(-\frac{\sqrt{213}}{6}-\frac{5}{2}\right)\right)\left(x-\left(\frac{\sqrt{213}}{6}-\frac{5}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$factor(3x^{2}+20x+25-8x+3x-24)$$

Combine 20x and -8x to get 12x.

$$factor(3x^{2}+12x+25+3x-24)$$

Combine 12x and 3x to get 15x.

$$factor(3x^{2}+15x+25-24)$$

Subtract 24 from 25 to get 1.

$$factor(3x^{2}+15x+1)$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$3x^{2}+15x+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-15±\sqrt{15^{2}-4\times 3}}{2\times 3}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-15±\sqrt{225-12}}{2\times 3}$$

Add 225 to -12.

$$x=\frac{-15±\sqrt{213}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-15±\sqrt{213}}{6}$$

Now solve the equation x=\frac{-15±\sqrt{213}}{6} when ± is plus. Add -15 to \sqrt{213}.

$$x=\frac{\sqrt{213}-15}{6}$$

Divide -15+\sqrt{213} by 6.

$$x=\frac{\sqrt{213}}{6}-\frac{5}{2}$$

Now solve the equation x=\frac{-15±\sqrt{213}}{6} when ± is minus. Subtract \sqrt{213} from -15.

$$x=\frac{-\sqrt{213}-15}{6}$$

Divide -15-\sqrt{213} by 6.

$$x=-\frac{\sqrt{213}}{6}-\frac{5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{2}+\frac{\sqrt{213}}{6} for x_{1} and -\frac{5}{2}-\frac{\sqrt{213}}{6} for x_{2}.

$$3x^{2}+15x+1=3\left(x-\left(\frac{\sqrt{213}}{6}-\frac{5}{2}\right)\right)\left(x-\left(-\frac{\sqrt{213}}{6}-\frac{5}{2}\right)\right)$$