Solve 4x^2+6x-13

$$4 { x }^{ 2 } +6x-13$$

Factor

$$4\left(x-\frac{-\sqrt{61}-3}{4}\right)\left(x-\frac{\sqrt{61}-3}{4}\right)$$

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.

$$4x^{2}+6x-13=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\times 4\left(-13\right)}}{2\times 4}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 4\left(-13\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-6±\sqrt{36-16\left(-13\right)}}{2\times 4}$$

Multiply -16 times -13.

$$x=\frac{-6±\sqrt{36+208}}{2\times 4}$$

Add 36 to 208.

$$x=\frac{-6±\sqrt{244}}{2\times 4}$$

Take the square root of 244.

$$x=\frac{-6±2\sqrt{61}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-6±2\sqrt{61}}{8}$$

Now solve the equation x=\frac{-6±2\sqrt{61}}{8} when ± is plus. Add -6 to 2\sqrt{61}.

$$x=\frac{2\sqrt{61}-6}{8}$$

Divide -6+2\sqrt{61} by 8.

$$x=\frac{\sqrt{61}-3}{4}$$

Now solve the equation x=\frac{-6±2\sqrt{61}}{8} when ± is minus. Subtract 2\sqrt{61} from -6.

$$x=\frac{-2\sqrt{61}-6}{8}$$

Divide -6-2\sqrt{61} by 8.

$$x=\frac{-\sqrt{61}-3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+\sqrt{61}}{4} for x_{1} and \frac{-3-\sqrt{61}}{4} for x_{2}.

$$4x^{2}+6x-13=4\left(x-\frac{\sqrt{61}-3}{4}\right)\left(x-\frac{-\sqrt{61}-3}{4}\right)$$

Evaluate

$$4x^{2}+6x-13$$