Solve 5x^2-26x-3

$$5 { x }^{ 2 } -26x-3$$

Factor

$$5\left(x-\frac{13-2\sqrt{46}}{5}\right)\left(x-\frac{2\sqrt{46}+13}{5}\right)$$

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.

$$5x^{2}-26x-3=0$$

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.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 5\left(-3\right)}}{2\times 5}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 5\left(-3\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-26\right)±\sqrt{676-20\left(-3\right)}}{2\times 5}$$

Multiply -20 times -3.

$$x=\frac{-\left(-26\right)±\sqrt{676+60}}{2\times 5}$$

Add 676 to 60.

$$x=\frac{-\left(-26\right)±\sqrt{736}}{2\times 5}$$

Take the square root of 736.

$$x=\frac{-\left(-26\right)±4\sqrt{46}}{2\times 5}$$

The opposite of -26 is 26.

$$x=\frac{26±4\sqrt{46}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{26±4\sqrt{46}}{10}$$

Now solve the equation x=\frac{26±4\sqrt{46}}{10} when ± is plus. Add 26 to 4\sqrt{46}.

$$x=\frac{4\sqrt{46}+26}{10}$$

Divide 26+4\sqrt{46} by 10.

$$x=\frac{2\sqrt{46}+13}{5}$$

Now solve the equation x=\frac{26±4\sqrt{46}}{10} when ± is minus. Subtract 4\sqrt{46} from 26.

$$x=\frac{26-4\sqrt{46}}{10}$$

Divide 26-4\sqrt{46} by 10.

$$x=\frac{13-2\sqrt{46}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+2\sqrt{46}}{5} for x_{1} and \frac{13-2\sqrt{46}}{5} for x_{2}.

$$5x^{2}-26x-3=5\left(x-\frac{2\sqrt{46}+13}{5}\right)\left(x-\frac{13-2\sqrt{46}}{5}\right)$$

Evaluate

$$5x^{2}-26x-3$$