Solve y(x)=-16x^2+48x+288

$$y(x)=-16 { x }^{ 2 } +48x+288$$

Factor

$$-16\left(x-6\right)\left(x+3\right)$$

Solution Steps

Factor out 16.

$$16\left(-x^{2}+3x+18\right)$$

Consider -x^{2}+3x+18. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=3 ab=-18=-18$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 3.

$$a=6 b=-3$$

Rewrite -x^{2}+3x+18 as \left(-x^{2}+6x\right)+\left(-3x+18\right).

$$\left(-x^{2}+6x\right)+\left(-3x+18\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(x-6\right)-3\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x-3\right)$$

Rewrite the complete factored expression.

$$16\left(x-6\right)\left(-x-3\right)$$

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.

$$-16x^{2}+48x+288=0$$

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.

$$x=\frac{-48±\sqrt{48^{2}-4\left(-16\right)\times 288}}{2\left(-16\right)}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\left(-16\right)\times 288}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-48±\sqrt{2304+64\times 288}}{2\left(-16\right)}$$

Multiply 64 times 288.

$$x=\frac{-48±\sqrt{2304+18432}}{2\left(-16\right)}$$

Add 2304 to 18432.

$$x=\frac{-48±\sqrt{20736}}{2\left(-16\right)}$$

Take the square root of 20736.

$$x=\frac{-48±144}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-48±144}{-32}$$

Now solve the equation x=\frac{-48±144}{-32} when ± is plus. Add -48 to 144.

$$x=\frac{96}{-32}$$

Divide 96 by -32.

$$x=-3$$

Now solve the equation x=\frac{-48±144}{-32} when ± is minus. Subtract 144 from -48.

$$x=-\frac{192}{-32}$$

Divide -192 by -32.

$$x=6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and 6 for x_{2}.

$$-16x^{2}+48x+288=-16\left(x-\left(-3\right)\right)\left(x-6\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-16x^{2}+48x+288=-16\left(x+3\right)\left(x-6\right)$$

Evaluate

$$-16\left(x-6\right)\left(x+3\right)$$