Solve 14x^2-141x+90

$$14 { x }^{ 2 } -141x+90$$

Factor

$$14\left(x-\frac{141-3\sqrt{1649}}{28}\right)\left(x-\frac{3\sqrt{1649}+141}{28}\right)$$

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.

$$14x^{2}-141x+90=0$$

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.

$$x=\frac{-\left(-141\right)±\sqrt{\left(-141\right)^{2}-4\times 14\times 90}}{2\times 14}$$

Square -141.

$$x=\frac{-\left(-141\right)±\sqrt{19881-4\times 14\times 90}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-\left(-141\right)±\sqrt{19881-56\times 90}}{2\times 14}$$

Multiply -56 times 90.

$$x=\frac{-\left(-141\right)±\sqrt{19881-5040}}{2\times 14}$$

Add 19881 to -5040.

$$x=\frac{-\left(-141\right)±\sqrt{14841}}{2\times 14}$$

Take the square root of 14841.

$$x=\frac{-\left(-141\right)±3\sqrt{1649}}{2\times 14}$$

The opposite of -141 is 141.

$$x=\frac{141±3\sqrt{1649}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{141±3\sqrt{1649}}{28}$$

Now solve the equation x=\frac{141±3\sqrt{1649}}{28} when ± is plus. Add 141 to 3\sqrt{1649}.

$$x=\frac{3\sqrt{1649}+141}{28}$$

Now solve the equation x=\frac{141±3\sqrt{1649}}{28} when ± is minus. Subtract 3\sqrt{1649} from 141.

$$x=\frac{141-3\sqrt{1649}}{28}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{141+3\sqrt{1649}}{28} for x_{1} and \frac{141-3\sqrt{1649}}{28} for x_{2}.

$$14x^{2}-141x+90=14\left(x-\frac{3\sqrt{1649}+141}{28}\right)\left(x-\frac{141-3\sqrt{1649}}{28}\right)$$

Evaluate

$$14x^{2}-141x+90$$