Solve 25x^2+30x-9

$$25 { x }^{ 2 } +30x-9$$

Factor

$$25\left(x-\frac{-3\sqrt{2}-3}{5}\right)\left(x-\frac{3\sqrt{2}-3}{5}\right)$$

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.

$$25x^{2}+30x-9=0$$

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.

$$x=\frac{-30±\sqrt{30^{2}-4\times 25\left(-9\right)}}{2\times 25}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 25\left(-9\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-30±\sqrt{900-100\left(-9\right)}}{2\times 25}$$

Multiply -100 times -9.

$$x=\frac{-30±\sqrt{900+900}}{2\times 25}$$

Add 900 to 900.

$$x=\frac{-30±\sqrt{1800}}{2\times 25}$$

Take the square root of 1800.

$$x=\frac{-30±30\sqrt{2}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-30±30\sqrt{2}}{50}$$

Now solve the equation x=\frac{-30±30\sqrt{2}}{50} when ± is plus. Add -30 to 30\sqrt{2}.

$$x=\frac{30\sqrt{2}-30}{50}$$

Divide -30+30\sqrt{2} by 50.

$$x=\frac{3\sqrt{2}-3}{5}$$

Now solve the equation x=\frac{-30±30\sqrt{2}}{50} when ± is minus. Subtract 30\sqrt{2} from -30.

$$x=\frac{-30\sqrt{2}-30}{50}$$

Divide -30-30\sqrt{2} by 50.

$$x=\frac{-3\sqrt{2}-3}{5}$$

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

$$25x^{2}+30x-9=25\left(x-\frac{3\sqrt{2}-3}{5}\right)\left(x-\frac{-3\sqrt{2}-3}{5}\right)$$

Evaluate

$$25x^{2}+30x-9$$