Solve 25x^2-144x-64=

$$25 { x }^{ 2 } -144x-64=$$

Factor

$$25\left(x-\frac{72-8\sqrt{106}}{25}\right)\left(x-\frac{8\sqrt{106}+72}{25}\right)$$

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.

$$25x^{2}-144x-64=0$$

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.

$$x=\frac{-\left(-144\right)±\sqrt{\left(-144\right)^{2}-4\times 25\left(-64\right)}}{2\times 25}$$

Square -144.

$$x=\frac{-\left(-144\right)±\sqrt{20736-4\times 25\left(-64\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-144\right)±\sqrt{20736-100\left(-64\right)}}{2\times 25}$$

Multiply -100 times -64.

$$x=\frac{-\left(-144\right)±\sqrt{20736+6400}}{2\times 25}$$

Add 20736 to 6400.

$$x=\frac{-\left(-144\right)±\sqrt{27136}}{2\times 25}$$

Take the square root of 27136.

$$x=\frac{-\left(-144\right)±16\sqrt{106}}{2\times 25}$$

The opposite of -144 is 144.

$$x=\frac{144±16\sqrt{106}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{144±16\sqrt{106}}{50}$$

Now solve the equation x=\frac{144±16\sqrt{106}}{50} when ± is plus. Add 144 to 16\sqrt{106}.

$$x=\frac{16\sqrt{106}+144}{50}$$

Divide 144+16\sqrt{106} by 50.

$$x=\frac{8\sqrt{106}+72}{25}$$

Now solve the equation x=\frac{144±16\sqrt{106}}{50} when ± is minus. Subtract 16\sqrt{106} from 144.

$$x=\frac{144-16\sqrt{106}}{50}$$

Divide 144-16\sqrt{106} by 50.

$$x=\frac{72-8\sqrt{106}}{25}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{72+8\sqrt{106}}{25} for x_{1} and \frac{72-8\sqrt{106}}{25} for x_{2}.

$$25x^{2}-144x-64=25\left(x-\frac{8\sqrt{106}+72}{25}\right)\left(x-\frac{72-8\sqrt{106}}{25}\right)$$

Evaluate

$$25x^{2}-144x-64$$