Solve 7x^2-2x-70

$$7 { x }^{ 2 } -2x-70$$

Factor

$$7\left(x-\frac{1-\sqrt{491}}{7}\right)\left(x-\frac{\sqrt{491}+1}{7}\right)$$

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.

$$7x^{2}-2x-70=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 7\left(-70\right)}}{2\times 7}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 7\left(-70\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-2\right)±\sqrt{4-28\left(-70\right)}}{2\times 7}$$

Multiply -28 times -70.

$$x=\frac{-\left(-2\right)±\sqrt{4+1960}}{2\times 7}$$

Add 4 to 1960.

$$x=\frac{-\left(-2\right)±\sqrt{1964}}{2\times 7}$$

Take the square root of 1964.

$$x=\frac{-\left(-2\right)±2\sqrt{491}}{2\times 7}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{491}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{2±2\sqrt{491}}{14}$$

Now solve the equation x=\frac{2±2\sqrt{491}}{14} when ± is plus. Add 2 to 2\sqrt{491}.

$$x=\frac{2\sqrt{491}+2}{14}$$

Divide 2+2\sqrt{491} by 14.

$$x=\frac{\sqrt{491}+1}{7}$$

Now solve the equation x=\frac{2±2\sqrt{491}}{14} when ± is minus. Subtract 2\sqrt{491} from 2.

$$x=\frac{2-2\sqrt{491}}{14}$$

Divide 2-2\sqrt{491} by 14.

$$x=\frac{1-\sqrt{491}}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{491}}{7} for x_{1} and \frac{1-\sqrt{491}}{7} for x_{2}.

$$7x^{2}-2x-70=7\left(x-\frac{\sqrt{491}+1}{7}\right)\left(x-\frac{1-\sqrt{491}}{7}\right)$$

Evaluate

$$7x^{2}-2x-70$$