Solve -x^2+16x-28

$$- x ^ { 2 } + 16 x - 28$$

Factor

$$\left(2-x\right)\left(x-14\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-28. To find a and b, set up a system to be solved.

$$a+b=16 ab=-\left(-28\right)=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 16.

$$a=14 b=2$$

Rewrite -x^{2}+16x-28 as \left(-x^{2}+14x\right)+\left(2x-28\right).

$$\left(-x^{2}+14x\right)+\left(2x-28\right)$$

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

$$-x\left(x-14\right)+2\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(-x+2\right)$$

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.

$$-x^{2}+16x-28=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-16±\sqrt{256+4\left(-28\right)}}{2\left(-1\right)}$$

Multiply 4 times -28.

$$x=\frac{-16±\sqrt{256-112}}{2\left(-1\right)}$$

Add 256 to -112.

$$x=\frac{-16±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$x=\frac{-16±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-16±12}{-2}$$

Now solve the equation x=\frac{-16±12}{-2} when ± is plus. Add -16 to 12.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-16±12}{-2} when ± is minus. Subtract 12 from -16.

$$x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x=14$$

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 14 for x_{2}.

$$-x^{2}+16x-28=-\left(x-2\right)\left(x-14\right)$$

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.

$$x ^ 2 -16x +28 = 0$$

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

$$r + s = 16 rs = 28$$

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 = 28

$$(8 - u) (8 + u) = 28$$

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

$$64 - u^2 = 28$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = 28-64 = -36$$

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

$$u^2 = 36 u = \pm\sqrt{36} = \pm 6$$

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

$$r =8 - 6 = 2 s = 8 + 6 = 14$$

Evaluate

$$\left(2-x\right)\left(x-14\right)$$