Solve 25x^2+12x-7

$$25 { x }^{ 2 } +12x-7$$

Factor

$$25\left(x-\frac{-\sqrt{211}-6}{25}\right)\left(x-\frac{\sqrt{211}-6}{25}\right)$$

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.

$$25x^{2}+12x-7=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 25\left(-7\right)}}{2\times 25}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 25\left(-7\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-12±\sqrt{144-100\left(-7\right)}}{2\times 25}$$

Multiply -100 times -7.

$$x=\frac{-12±\sqrt{144+700}}{2\times 25}$$

Add 144 to 700.

$$x=\frac{-12±\sqrt{844}}{2\times 25}$$

Take the square root of 844.

$$x=\frac{-12±2\sqrt{211}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-12±2\sqrt{211}}{50}$$

Now solve the equation x=\frac{-12±2\sqrt{211}}{50} when ± is plus. Add -12 to 2\sqrt{211}.

$$x=\frac{2\sqrt{211}-12}{50}$$

Divide -12+2\sqrt{211} by 50.

$$x=\frac{\sqrt{211}-6}{25}$$

Now solve the equation x=\frac{-12±2\sqrt{211}}{50} when ± is minus. Subtract 2\sqrt{211} from -12.

$$x=\frac{-2\sqrt{211}-12}{50}$$

Divide -12-2\sqrt{211} by 50.

$$x=\frac{-\sqrt{211}-6}{25}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-6+\sqrt{211}}{25} for x_{1} and \frac{-6-\sqrt{211}}{25} for x_{2}.

$$25x^{2}+12x-7=25\left(x-\frac{\sqrt{211}-6}{25}\right)\left(x-\frac{-\sqrt{211}-6}{25}\right)$$

Evaluate

$$25x^{2}+12x-7$$