Solve 5x^2-21x-10

$$5 { x }^{ 2 } -21x-10$$

Factor

$$5\left(x-\frac{21-\sqrt{641}}{10}\right)\left(x-\frac{\sqrt{641}+21}{10}\right)$$

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.

$$5x^{2}-21x-10=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 5\left(-10\right)}}{2\times 5}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 5\left(-10\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-21\right)±\sqrt{441-20\left(-10\right)}}{2\times 5}$$

Multiply -20 times -10.

$$x=\frac{-\left(-21\right)±\sqrt{441+200}}{2\times 5}$$

Add 441 to 200.

$$x=\frac{-\left(-21\right)±\sqrt{641}}{2\times 5}$$

The opposite of -21 is 21.

$$x=\frac{21±\sqrt{641}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{21±\sqrt{641}}{10}$$

Now solve the equation x=\frac{21±\sqrt{641}}{10} when ± is plus. Add 21 to \sqrt{641}.

$$x=\frac{\sqrt{641}+21}{10}$$

Now solve the equation x=\frac{21±\sqrt{641}}{10} when ± is minus. Subtract \sqrt{641} from 21.

$$x=\frac{21-\sqrt{641}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{21+\sqrt{641}}{10} for x_{1} and \frac{21-\sqrt{641}}{10} for x_{2}.

$$5x^{2}-21x-10=5\left(x-\frac{\sqrt{641}+21}{10}\right)\left(x-\frac{21-\sqrt{641}}{10}\right)$$

Evaluate

$$5x^{2}-21x-10$$