Solve 4x-2x^2+12x-4x^2+5

$$4x-2 { x }^{ 2 } +12x-4 { x }^{ 2 } +5$$

Evaluate

$$5+16x-6x^{2}$$

Solution Steps

Combine 4x and 12x to get 16x.

$$16x-2x^{2}-4x^{2}+5$$

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

$$16x-6x^{2}+5$$

Factor

$$-6\left(x-\left(-\frac{\sqrt{94}}{6}+\frac{4}{3}\right)\right)\left(x-\left(\frac{\sqrt{94}}{6}+\frac{4}{3}\right)\right)$$

Steps Using the Quadratic Formula

Combine 4x and 12x to get 16x.

$$factor(16x-2x^{2}-4x^{2}+5)$$

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

$$factor(16x-6x^{2}+5)$$

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.

$$-6x^{2}+16x+5=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-6\right)\times 5}}{2\left(-6\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-6\right)\times 5}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-16±\sqrt{256+24\times 5}}{2\left(-6\right)}$$

Multiply 24 times 5.

$$x=\frac{-16±\sqrt{256+120}}{2\left(-6\right)}$$

Add 256 to 120.

$$x=\frac{-16±\sqrt{376}}{2\left(-6\right)}$$

Take the square root of 376.

$$x=\frac{-16±2\sqrt{94}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-16±2\sqrt{94}}{-12}$$

Now solve the equation x=\frac{-16±2\sqrt{94}}{-12} when ± is plus. Add -16 to 2\sqrt{94}.

$$x=\frac{2\sqrt{94}-16}{-12}$$

Divide -16+2\sqrt{94} by -12.

$$x=-\frac{\sqrt{94}}{6}+\frac{4}{3}$$

Now solve the equation x=\frac{-16±2\sqrt{94}}{-12} when ± is minus. Subtract 2\sqrt{94} from -16.

$$x=\frac{-2\sqrt{94}-16}{-12}$$

Divide -16-2\sqrt{94} by -12.

$$x=\frac{\sqrt{94}}{6}+\frac{4}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4}{3}-\frac{\sqrt{94}}{6} for x_{1} and \frac{4}{3}+\frac{\sqrt{94}}{6} for x_{2}.

$$-6x^{2}+16x+5=-6\left(x-\left(-\frac{\sqrt{94}}{6}+\frac{4}{3}\right)\right)\left(x-\left(\frac{\sqrt{94}}{6}+\frac{4}{3}\right)\right)$$