Solve 4x^2+7x-19

$$4 { x }^{ 2 } +7x-19$$

Factor

$$4\left(x-\frac{-\sqrt{353}-7}{8}\right)\left(x-\frac{\sqrt{353}-7}{8}\right)$$

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.

$$4x^{2}+7x-19=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 4\left(-19\right)}}{2\times 4}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 4\left(-19\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-7±\sqrt{49-16\left(-19\right)}}{2\times 4}$$

Multiply -16 times -19.

$$x=\frac{-7±\sqrt{49+304}}{2\times 4}$$

Add 49 to 304.

$$x=\frac{-7±\sqrt{353}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-7±\sqrt{353}}{8}$$

Now solve the equation x=\frac{-7±\sqrt{353}}{8} when ± is plus. Add -7 to \sqrt{353}.

$$x=\frac{\sqrt{353}-7}{8}$$

Now solve the equation x=\frac{-7±\sqrt{353}}{8} when ± is minus. Subtract \sqrt{353} from -7.

$$x=\frac{-\sqrt{353}-7}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{353}}{8} for x_{1} and \frac{-7-\sqrt{353}}{8} for x_{2}.

$$4x^{2}+7x-19=4\left(x-\frac{\sqrt{353}-7}{8}\right)\left(x-\frac{-\sqrt{353}-7}{8}\right)$$

Evaluate

$$4x^{2}+7x-19$$