Solve 2-2x^2+14x+27

$$2-2 { x }^{ 2 } +14x+27$$

Factor

$$-2\left(x-\frac{7-\sqrt{107}}{2}\right)\left(x-\frac{\sqrt{107}+7}{2}\right)$$

Steps Using the Quadratic Formula

Add 2 and 27 to get 29.

$$factor(29-2x^{2}+14x)$$

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.

$$-2x^{2}+14x+29=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-2\right)\times 29}}{2\left(-2\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-2\right)\times 29}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-14±\sqrt{196+8\times 29}}{2\left(-2\right)}$$

Multiply 8 times 29.

$$x=\frac{-14±\sqrt{196+232}}{2\left(-2\right)}$$

Add 196 to 232.

$$x=\frac{-14±\sqrt{428}}{2\left(-2\right)}$$

Take the square root of 428.

$$x=\frac{-14±2\sqrt{107}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-14±2\sqrt{107}}{-4}$$

Now solve the equation x=\frac{-14±2\sqrt{107}}{-4} when ± is plus. Add -14 to 2\sqrt{107}.

$$x=\frac{2\sqrt{107}-14}{-4}$$

Divide -14+2\sqrt{107} by -4.

$$x=\frac{7-\sqrt{107}}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{107}}{-4} when ± is minus. Subtract 2\sqrt{107} from -14.

$$x=\frac{-2\sqrt{107}-14}{-4}$$

Divide -14-2\sqrt{107} by -4.

$$x=\frac{\sqrt{107}+7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{107}}{2} for x_{1} and \frac{7+\sqrt{107}}{2} for x_{2}.

$$-2x^{2}+14x+29=-2\left(x-\frac{7-\sqrt{107}}{2}\right)\left(x-\frac{\sqrt{107}+7}{2}\right)$$

Evaluate

$$29+14x-2x^{2}$$

Solution Steps

Add 2 and 27 to get 29.

$$29-2x^{2}+14x$$