Solve 4a^2-19a+7

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

Factor

$$4\left(a-\frac{19-\sqrt{249}}{8}\right)\left(a-\frac{\sqrt{249}+19}{8}\right)$$

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.

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

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.

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

Square -19.

$$a=\frac{-\left(-19\right)±\sqrt{361-4\times 4\times 7}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-19\right)±\sqrt{361-16\times 7}}{2\times 4}$$

Multiply -16 times 7.

$$a=\frac{-\left(-19\right)±\sqrt{361-112}}{2\times 4}$$

Add 361 to -112.

$$a=\frac{-\left(-19\right)±\sqrt{249}}{2\times 4}$$

The opposite of -19 is 19.

$$a=\frac{19±\sqrt{249}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{19±\sqrt{249}}{8}$$

Now solve the equation a=\frac{19±\sqrt{249}}{8} when ± is plus. Add 19 to \sqrt{249}.

$$a=\frac{\sqrt{249}+19}{8}$$

Now solve the equation a=\frac{19±\sqrt{249}}{8} when ± is minus. Subtract \sqrt{249} from 19.

$$a=\frac{19-\sqrt{249}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19+\sqrt{249}}{8} for x_{1} and \frac{19-\sqrt{249}}{8} for x_{2}.

$$4a^{2}-19a+7=4\left(a-\frac{\sqrt{249}+19}{8}\right)\left(a-\frac{19-\sqrt{249}}{8}\right)$$

Evaluate

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