Solve 70x^2-13500x-2550000

$$70 { x }^{ 2 } -13500x-2550000$$

Factor

$$70\left(x-\frac{675-25\sqrt{3585}}{7}\right)\left(x-\frac{25\sqrt{3585}+675}{7}\right)$$

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.

$$70x^{2}-13500x-2550000=0$$

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.

$$x=\frac{-\left(-13500\right)±\sqrt{\left(-13500\right)^{2}-4\times 70\left(-2550000\right)}}{2\times 70}$$

Square -13500.

$$x=\frac{-\left(-13500\right)±\sqrt{182250000-4\times 70\left(-2550000\right)}}{2\times 70}$$

Multiply -4 times 70.

$$x=\frac{-\left(-13500\right)±\sqrt{182250000-280\left(-2550000\right)}}{2\times 70}$$

Multiply -280 times -2550000.

$$x=\frac{-\left(-13500\right)±\sqrt{182250000+714000000}}{2\times 70}$$

Add 182250000 to 714000000.

$$x=\frac{-\left(-13500\right)±\sqrt{896250000}}{2\times 70}$$

Take the square root of 896250000.

$$x=\frac{-\left(-13500\right)±500\sqrt{3585}}{2\times 70}$$

The opposite of -13500 is 13500.

$$x=\frac{13500±500\sqrt{3585}}{2\times 70}$$

Multiply 2 times 70.

$$x=\frac{13500±500\sqrt{3585}}{140}$$

Now solve the equation x=\frac{13500±500\sqrt{3585}}{140} when ± is plus. Add 13500 to 500\sqrt{3585}.

$$x=\frac{500\sqrt{3585}+13500}{140}$$

Divide 13500+500\sqrt{3585} by 140.

$$x=\frac{25\sqrt{3585}+675}{7}$$

Now solve the equation x=\frac{13500±500\sqrt{3585}}{140} when ± is minus. Subtract 500\sqrt{3585} from 13500.

$$x=\frac{13500-500\sqrt{3585}}{140}$$

Divide 13500-500\sqrt{3585} by 140.

$$x=\frac{675-25\sqrt{3585}}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{675+25\sqrt{3585}}{7} for x_{1} and \frac{675-25\sqrt{3585}}{7} for x_{2}.

$$70x^{2}-13500x-2550000=70\left(x-\frac{25\sqrt{3585}+675}{7}\right)\left(x-\frac{675-25\sqrt{3585}}{7}\right)$$

Evaluate

$$70x^{2}-13500x-2550000$$