Solve f(x)=-2x^2+16x-24

$$f ( x ) = - 2 x ^ { 2 } + 16 x - 24$$

Factor

$$2\left(2-x\right)\left(x-6\right)$$

Solution Steps

Factor out 2.

$$2\left(-x^{2}+8x-12\right)$$

Consider -x^{2}+8x-12. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-12\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=6 b=2$$

Rewrite -x^{2}+8x-12 as \left(-x^{2}+6x\right)+\left(2x-12\right).

$$\left(-x^{2}+6x\right)+\left(2x-12\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-6\right)+2\left(x-6\right)$$

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

$$\left(x-6\right)\left(-x+2\right)$$

Rewrite the complete factored expression.

$$2\left(x-6\right)\left(-x+2\right)$$

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.

$$-2x^{2}+16x-24=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-2\right)\left(-24\right)}}{2\left(-2\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-2\right)\left(-24\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-16±\sqrt{256+8\left(-24\right)}}{2\left(-2\right)}$$

Multiply 8 times -24.

$$x=\frac{-16±\sqrt{256-192}}{2\left(-2\right)}$$

Add 256 to -192.

$$x=\frac{-16±\sqrt{64}}{2\left(-2\right)}$$

Take the square root of 64.

$$x=\frac{-16±8}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-16±8}{-4}$$

Now solve the equation x=\frac{-16±8}{-4} when ± is plus. Add -16 to 8.

$$x=-\frac{8}{-4}$$

Divide -8 by -4.

$$x=2$$

Now solve the equation x=\frac{-16±8}{-4} when ± is minus. Subtract 8 from -16.

$$x=-\frac{24}{-4}$$

Divide -24 by -4.

$$x=6$$

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

$$-2x^{2}+16x-24=-2\left(x-2\right)\left(x-6\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -8x +12 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 8 rs = 12$$

Two numbers r and s sum up to 8 exactly when the average of the two numbers is \frac{1}{2}*8 = 4. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 4 - u s = 4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 12

$$(4 - u) (4 + u) = 12$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$16 - u^2 = 12$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = 12-16 = -4$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 4 u = \pm\sqrt{4} = \pm 2$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =4 - 2 = 2 s = 4 + 2 = 6$$

Evaluate

$$2\left(2-x\right)\left(x-6\right)$$