Solve f(x)=x^2-16x+48

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

Factor

$$\left(x-12\right)\left(x-4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+48. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 48=48$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Calculate the sum for each pair.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

The solution is the pair that gives sum -16.

$$a=-12 b=-4$$

Rewrite x^{2}-16x+48 as \left(x^{2}-12x\right)+\left(-4x+48\right).

$$\left(x^{2}-12x\right)+\left(-4x+48\right)$$

Factor out x in the first and -4 in the second group.

$$x\left(x-12\right)-4\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x-4\right)$$

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.

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

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.

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

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 48}}{2}$$

Multiply -4 times 48.

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

Add 256 to -192.

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

Take the square root of 64.

$$x=\frac{-\left(-16\right)±8}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±8}{2}$$

Now solve the equation x=\frac{16±8}{2} when ± is plus. Add 16 to 8.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{16±8}{2} when ± is minus. Subtract 8 from 16.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12 for x_{1} and 4 for x_{2}.

$$x^{2}-16x+48=\left(x-12\right)\left(x-4\right)$$

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.

$$x ^ 2 -16x +48 = 0$$

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

$$r + s = 16 rs = 48$$

Two numbers r and s sum up to 16 exactly when the average of the two numbers is \frac{1}{2}*16 = 8. 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>

$$r = 8 - u s = 8 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 48

$$(8 - u) (8 + u) = 48$$

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

$$64 - u^2 = 48$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = 48-64 = -16$$

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

$$u^2 = 16 u = \pm\sqrt{16} = \pm 4$$

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

$$r =8 - 4 = 4 s = 8 + 4 = 12$$

Evaluate

$$\left(x-12\right)\left(x-4\right)$$