Solve 10x^2-5x-70

$$10 { x }^{ 2 } -5x-70$$

Factor

$$10\left(x-\frac{1-\sqrt{113}}{4}\right)\left(x-\frac{\sqrt{113}+1}{4}\right)$$

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.

$$10x^{2}-5x-70=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 10\left(-70\right)}}{2\times 10}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 10\left(-70\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-5\right)±\sqrt{25-40\left(-70\right)}}{2\times 10}$$

Multiply -40 times -70.

$$x=\frac{-\left(-5\right)±\sqrt{25+2800}}{2\times 10}$$

Add 25 to 2800.

$$x=\frac{-\left(-5\right)±\sqrt{2825}}{2\times 10}$$

Take the square root of 2825.

$$x=\frac{-\left(-5\right)±5\sqrt{113}}{2\times 10}$$

The opposite of -5 is 5.

$$x=\frac{5±5\sqrt{113}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{5±5\sqrt{113}}{20}$$

Now solve the equation x=\frac{5±5\sqrt{113}}{20} when ± is plus. Add 5 to 5\sqrt{113}.

$$x=\frac{5\sqrt{113}+5}{20}$$

Divide 5+5\sqrt{113} by 20.

$$x=\frac{\sqrt{113}+1}{4}$$

Now solve the equation x=\frac{5±5\sqrt{113}}{20} when ± is minus. Subtract 5\sqrt{113} from 5.

$$x=\frac{5-5\sqrt{113}}{20}$$

Divide 5-5\sqrt{113} by 20.

$$x=\frac{1-\sqrt{113}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{113}}{4} for x_{1} and \frac{1-\sqrt{113}}{4} for x_{2}.

$$10x^{2}-5x-70=10\left(x-\frac{\sqrt{113}+1}{4}\right)\left(x-\frac{1-\sqrt{113}}{4}\right)$$

Evaluate

$$10x^{2}-5x-70$$