Solve -3a^2+31a+28

$$-3 { a }^{ 2 } +31a+28$$

Factor

$$-3\left(a-\frac{31-\sqrt{1297}}{6}\right)\left(a-\frac{\sqrt{1297}+31}{6}\right)$$

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.

$$-3a^{2}+31a+28=0$$

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.

$$a=\frac{-31±\sqrt{31^{2}-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Square 31.

$$a=\frac{-31±\sqrt{961-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$a=\frac{-31±\sqrt{961+12\times 28}}{2\left(-3\right)}$$

Multiply 12 times 28.

$$a=\frac{-31±\sqrt{961+336}}{2\left(-3\right)}$$

Add 961 to 336.

$$a=\frac{-31±\sqrt{1297}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$a=\frac{-31±\sqrt{1297}}{-6}$$

Now solve the equation a=\frac{-31±\sqrt{1297}}{-6} when ± is plus. Add -31 to \sqrt{1297}.

$$a=\frac{\sqrt{1297}-31}{-6}$$

Divide -31+\sqrt{1297} by -6.

$$a=\frac{31-\sqrt{1297}}{6}$$

Now solve the equation a=\frac{-31±\sqrt{1297}}{-6} when ± is minus. Subtract \sqrt{1297} from -31.

$$a=\frac{-\sqrt{1297}-31}{-6}$$

Divide -31-\sqrt{1297} by -6.

$$a=\frac{\sqrt{1297}+31}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{31-\sqrt{1297}}{6} for x_{1} and \frac{31+\sqrt{1297}}{6} for x_{2}.

$$-3a^{2}+31a+28=-3\left(a-\frac{31-\sqrt{1297}}{6}\right)\left(a-\frac{\sqrt{1297}+31}{6}\right)$$

Evaluate

$$28+31a-3a^{2}$$