Solve x^2-19x-66

$$x ^ { 2 } - 19 x - 66$$

Factor

$$\left(x-22\right)\left(x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-66. To find a and b, set up a system to be solved.

$$a+b=-19 ab=1\left(-66\right)=-66$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -66.

$$1,-66 2,-33 3,-22 6,-11$$

Calculate the sum for each pair.

$$1-66=-65 2-33=-31 3-22=-19 6-11=-5$$

The solution is the pair that gives sum -19.

$$a=-22 b=3$$

Rewrite x^{2}-19x-66 as \left(x^{2}-22x\right)+\left(3x-66\right).

$$\left(x^{2}-22x\right)+\left(3x-66\right)$$

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

$$x\left(x-22\right)+3\left(x-22\right)$$

Factor out common term x-22 by using distributive property.

$$\left(x-22\right)\left(x+3\right)$$

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.

$$x^{2}-19x-66=0$$

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.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\left(-66\right)}}{2}$$

Square -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\left(-66\right)}}{2}$$

Multiply -4 times -66.

$$x=\frac{-\left(-19\right)±\sqrt{361+264}}{2}$$

Add 361 to 264.

$$x=\frac{-\left(-19\right)±\sqrt{625}}{2}$$

Take the square root of 625.

$$x=\frac{-\left(-19\right)±25}{2}$$

The opposite of -19 is 19.

$$x=\frac{19±25}{2}$$

Now solve the equation x=\frac{19±25}{2} when ± is plus. Add 19 to 25.

$$x=\frac{44}{2}$$

Divide 44 by 2.

$$x=22$$

Now solve the equation x=\frac{19±25}{2} when ± is minus. Subtract 25 from 19.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 22 for x_{1} and -3 for x_{2}.

$$x^{2}-19x-66=\left(x-22\right)\left(x-\left(-3\right)\right)$$

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

$$x^{2}-19x-66=\left(x-22\right)\left(x+3\right)$$

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.

$$x ^ 2 -19x -66 = 0$$

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

$$r + s = 19 rs = -66$$

Two numbers r and s sum up to 19 exactly when the average of the two numbers is \frac{1}{2}*19 = \frac{19}{2}. 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>

$$r = \frac{19}{2} - u s = \frac{19}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -66

$$(\frac{19}{2} - u) (\frac{19}{2} + u) = -66$$

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

$$\frac{361}{4} - u^2 = -66$$

Simplify the expression by subtracting \frac{361}{4} on both sides

$$-u^2 = -66-\frac{361}{4} = -\frac{625}{4}$$

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

$$u^2 = \frac{625}{4} u = \pm\sqrt{\frac{625}{4}} = \pm \frac{25}{2}$$

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

$$r =\frac{19}{2} - \frac{25}{2} = -3 s = \frac{19}{2} + \frac{25}{2} = 22$$

Evaluate

$$\left(x-22\right)\left(x+3\right)$$