Solve 2x+8-3x^2+24

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

Factor

$$-3\left(x-\frac{1-\sqrt{97}}{3}\right)\left(x-\frac{\sqrt{97}+1}{3}\right)$$

Steps Using the Quadratic Formula

Add 8 and 24 to get 32.

$$factor(2x+32-3x^{2})$$

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.

$$-3x^{2}+2x+32=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-3\right)\times 32}}{2\left(-3\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-3\right)\times 32}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-2±\sqrt{4+12\times 32}}{2\left(-3\right)}$$

Multiply 12 times 32.

$$x=\frac{-2±\sqrt{4+384}}{2\left(-3\right)}$$

Add 4 to 384.

$$x=\frac{-2±\sqrt{388}}{2\left(-3\right)}$$

Take the square root of 388.

$$x=\frac{-2±2\sqrt{97}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-2±2\sqrt{97}}{-6}$$

Now solve the equation x=\frac{-2±2\sqrt{97}}{-6} when ± is plus. Add -2 to 2\sqrt{97}.

$$x=\frac{2\sqrt{97}-2}{-6}$$

Divide -2+2\sqrt{97} by -6.

$$x=\frac{1-\sqrt{97}}{3}$$

Now solve the equation x=\frac{-2±2\sqrt{97}}{-6} when ± is minus. Subtract 2\sqrt{97} from -2.

$$x=\frac{-2\sqrt{97}-2}{-6}$$

Divide -2-2\sqrt{97} by -6.

$$x=\frac{\sqrt{97}+1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{97}}{3} for x_{1} and \frac{1+\sqrt{97}}{3} for x_{2}.

$$-3x^{2}+2x+32=-3\left(x-\frac{1-\sqrt{97}}{3}\right)\left(x-\frac{\sqrt{97}+1}{3}\right)$$

Evaluate

$$32+2x-3x^{2}$$

Solution Steps

Add 8 and 24 to get 32.

$$2x+32-3x^{2}$$