Solve -3x^2+48x+5

$$-3 { x }^{ 2 } +48x+5$$

Factor

$$-3\left(x-\left(-\frac{\sqrt{591}}{3}+8\right)\right)\left(x-\left(\frac{\sqrt{591}}{3}+8\right)\right)$$

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.

$$-3x^{2}+48x+5=0$$

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.

$$x=\frac{-48±\sqrt{48^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-48±\sqrt{2304+12\times 5}}{2\left(-3\right)}$$

Multiply 12 times 5.

$$x=\frac{-48±\sqrt{2304+60}}{2\left(-3\right)}$$

Add 2304 to 60.

$$x=\frac{-48±\sqrt{2364}}{2\left(-3\right)}$$

Take the square root of 2364.

$$x=\frac{-48±2\sqrt{591}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-48±2\sqrt{591}}{-6}$$

Now solve the equation x=\frac{-48±2\sqrt{591}}{-6} when ± is plus. Add -48 to 2\sqrt{591}.

$$x=\frac{2\sqrt{591}-48}{-6}$$

Divide -48+2\sqrt{591} by -6.

$$x=-\frac{\sqrt{591}}{3}+8$$

Now solve the equation x=\frac{-48±2\sqrt{591}}{-6} when ± is minus. Subtract 2\sqrt{591} from -48.

$$x=\frac{-2\sqrt{591}-48}{-6}$$

Divide -48-2\sqrt{591} by -6.

$$x=\frac{\sqrt{591}}{3}+8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8-\frac{\sqrt{591}}{3} for x_{1} and 8+\frac{\sqrt{591}}{3} for x_{2}.

$$-3x^{2}+48x+5=-3\left(x-\left(-\frac{\sqrt{591}}{3}+8\right)\right)\left(x-\left(\frac{\sqrt{591}}{3}+8\right)\right)$$

Evaluate

$$5+48x-3x^{2}$$