Solve c^2-25c+66

$$c ^ { 2 } - 25 c + 66$$

Factor

$$\left(c-22\right)\left(c-3\right)$$

Solution Steps

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

$$a+b=-25 ab=1\times 66=66$$

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 66.

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

Calculate the sum for each pair.

$$-1-66=-67 -2-33=-35 -3-22=-25 -6-11=-17$$

The solution is the pair that gives sum -25.

$$a=-22 b=-3$$

Rewrite c^{2}-25c+66 as \left(c^{2}-22c\right)+\left(-3c+66\right).

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

Factor out c in the first and -3 in the second group.

$$c\left(c-22\right)-3\left(c-22\right)$$

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

$$\left(c-22\right)\left(c-3\right)$$

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.

$$c^{2}-25c+66=0$$

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.

$$c=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 66}}{2}$$

Square -25.

$$c=\frac{-\left(-25\right)±\sqrt{625-4\times 66}}{2}$$

Multiply -4 times 66.

$$c=\frac{-\left(-25\right)±\sqrt{625-264}}{2}$$

Add 625 to -264.

$$c=\frac{-\left(-25\right)±\sqrt{361}}{2}$$

Take the square root of 361.

$$c=\frac{-\left(-25\right)±19}{2}$$

The opposite of -25 is 25.

$$c=\frac{25±19}{2}$$

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

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

Divide 44 by 2.

$$c=22$$

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

$$c=\frac{6}{2}$$

Divide 6 by 2.

$$c=3$$

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}.

$$c^{2}-25c+66=\left(c-22\right)\left(c-3\right)$$

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.

$$x ^ 2 -25x +66 = 0$$

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

$$r + s = 25 rs = 66$$

Two numbers r and s sum up to 25 exactly when the average of the two numbers is \frac{1}{2}*25 = \frac{25}{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>

$$r = \frac{25}{2} - u s = \frac{25}{2} + u$$

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

$$(\frac{25}{2} - u) (\frac{25}{2} + u) = 66$$

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

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

Simplify the expression by subtracting \frac{625}{4} on both sides

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

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

$$u^2 = \frac{361}{4} u = \pm\sqrt{\frac{361}{4}} = \pm \frac{19}{2}$$

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

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

Evaluate

$$\left(c-22\right)\left(c-3\right)$$