Solve 6x^2+14x-28

$$6 { x }^{ 2 } +14x-28$$

Factor

$$6\left(x-\frac{-\sqrt{217}-7}{6}\right)\left(x-\frac{\sqrt{217}-7}{6}\right)$$

Steps Using the Quadratic Formula

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.

$$6x^{2}+14x-28=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\times 6\left(-28\right)}}{2\times 6}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 6\left(-28\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-14±\sqrt{196-24\left(-28\right)}}{2\times 6}$$

Multiply -24 times -28.

$$x=\frac{-14±\sqrt{196+672}}{2\times 6}$$

Add 196 to 672.

$$x=\frac{-14±\sqrt{868}}{2\times 6}$$

Take the square root of 868.

$$x=\frac{-14±2\sqrt{217}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-14±2\sqrt{217}}{12}$$

Now solve the equation x=\frac{-14±2\sqrt{217}}{12} when ± is plus. Add -14 to 2\sqrt{217}.

$$x=\frac{2\sqrt{217}-14}{12}$$

Divide -14+2\sqrt{217} by 12.

$$x=\frac{\sqrt{217}-7}{6}$$

Now solve the equation x=\frac{-14±2\sqrt{217}}{12} when ± is minus. Subtract 2\sqrt{217} from -14.

$$x=\frac{-2\sqrt{217}-14}{12}$$

Divide -14-2\sqrt{217} by 12.

$$x=\frac{-\sqrt{217}-7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{217}}{6} for x_{1} and \frac{-7-\sqrt{217}}{6} for x_{2}.

$$6x^{2}+14x-28=6\left(x-\frac{\sqrt{217}-7}{6}\right)\left(x-\frac{-\sqrt{217}-7}{6}\right)$$

Evaluate

$$6x^{2}+14x-28$$