Solve 12x^2-280x+1125

$$12 { x }^{ 2 } -280x+1125$$

Factor

$$12\left(x-\left(-\frac{5\sqrt{61}}{6}+\frac{35}{3}\right)\right)\left(x-\left(\frac{5\sqrt{61}}{6}+\frac{35}{3}\right)\right)$$

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.

$$12x^{2}-280x+1125=0$$

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.

$$x=\frac{-\left(-280\right)±\sqrt{\left(-280\right)^{2}-4\times 12\times 1125}}{2\times 12}$$

Square -280.

$$x=\frac{-\left(-280\right)±\sqrt{78400-4\times 12\times 1125}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-280\right)±\sqrt{78400-48\times 1125}}{2\times 12}$$

Multiply -48 times 1125.

$$x=\frac{-\left(-280\right)±\sqrt{78400-54000}}{2\times 12}$$

Add 78400 to -54000.

$$x=\frac{-\left(-280\right)±\sqrt{24400}}{2\times 12}$$

Take the square root of 24400.

$$x=\frac{-\left(-280\right)±20\sqrt{61}}{2\times 12}$$

The opposite of -280 is 280.

$$x=\frac{280±20\sqrt{61}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{280±20\sqrt{61}}{24}$$

Now solve the equation x=\frac{280±20\sqrt{61}}{24} when ± is plus. Add 280 to 20\sqrt{61}.

$$x=\frac{20\sqrt{61}+280}{24}$$

Divide 280+20\sqrt{61} by 24.

$$x=\frac{5\sqrt{61}}{6}+\frac{35}{3}$$

Now solve the equation x=\frac{280±20\sqrt{61}}{24} when ± is minus. Subtract 20\sqrt{61} from 280.

$$x=\frac{280-20\sqrt{61}}{24}$$

Divide 280-20\sqrt{61} by 24.

$$x=-\frac{5\sqrt{61}}{6}+\frac{35}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{35}{3}+\frac{5\sqrt{61}}{6} for x_{1} and \frac{35}{3}-\frac{5\sqrt{61}}{6} for x_{2}.

$$12x^{2}-280x+1125=12\left(x-\left(\frac{5\sqrt{61}}{6}+\frac{35}{3}\right)\right)\left(x-\left(-\frac{5\sqrt{61}}{6}+\frac{35}{3}\right)\right)$$

Evaluate

$$12x^{2}-280x+1125$$