Solve x^2-38x+192

$${ x }^{ 2 } -38x+192$$

Factor

$$\left(x-32\right)\left(x-6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+192. To find a and b, set up a system to be solved.

$$a+b=-38 ab=1\times 192=192$$

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 192.

$$-1,-192 -2,-96 -3,-64 -4,-48 -6,-32 -8,-24 -12,-16$$

Calculate the sum for each pair.

$$-1-192=-193 -2-96=-98 -3-64=-67 -4-48=-52 -6-32=-38 -8-24=-32 -12-16=-28$$

The solution is the pair that gives sum -38.

$$a=-32 b=-6$$

Rewrite x^{2}-38x+192 as \left(x^{2}-32x\right)+\left(-6x+192\right).

$$\left(x^{2}-32x\right)+\left(-6x+192\right)$$

Factor out x in the first and -6 in the second group.

$$x\left(x-32\right)-6\left(x-32\right)$$

Factor out common term x-32 by using distributive property.

$$\left(x-32\right)\left(x-6\right)$$

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.

$$x^{2}-38x+192=0$$

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.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\times 192}}{2}$$

Square -38.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\times 192}}{2}$$

Multiply -4 times 192.

$$x=\frac{-\left(-38\right)±\sqrt{1444-768}}{2}$$

Add 1444 to -768.

$$x=\frac{-\left(-38\right)±\sqrt{676}}{2}$$

Take the square root of 676.

$$x=\frac{-\left(-38\right)±26}{2}$$

The opposite of -38 is 38.

$$x=\frac{38±26}{2}$$

Now solve the equation x=\frac{38±26}{2} when ± is plus. Add 38 to 26.

$$x=\frac{64}{2}$$

Divide 64 by 2.

$$x=32$$

Now solve the equation x=\frac{38±26}{2} when ± is minus. Subtract 26 from 38.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 32 for x_{1} and 6 for x_{2}.

$$x^{2}-38x+192=\left(x-32\right)\left(x-6\right)$$

Evaluate

$$\left(x-32\right)\left(x-6\right)$$