Solve c^2-12c+27

$$c ^ { 2 } - 12 c + 27$$

Factor

$$\left(c-9\right)\left(c-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as c^{2}+ac+bc+27. To find a and b, set up a system to be solved.

$$a+b=-12 ab=1\times 27=27$$

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 27.

$$-1,-27 -3,-9$$

Calculate the sum for each pair.

$$-1-27=-28 -3-9=-12$$

The solution is the pair that gives sum -12.

$$a=-9 b=-3$$

Rewrite c^{2}-12c+27 as \left(c^{2}-9c\right)+\left(-3c+27\right).

$$\left(c^{2}-9c\right)+\left(-3c+27\right)$$

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

$$c\left(c-9\right)-3\left(c-9\right)$$

Factor out common term c-9 by using distributive property.

$$\left(c-9\right)\left(c-3\right)$$

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.

$$c^{2}-12c+27=0$$

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.

$$c=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 27}}{2}$$

Square -12.

$$c=\frac{-\left(-12\right)±\sqrt{144-4\times 27}}{2}$$

Multiply -4 times 27.

$$c=\frac{-\left(-12\right)±\sqrt{144-108}}{2}$$

Add 144 to -108.

$$c=\frac{-\left(-12\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$c=\frac{-\left(-12\right)±6}{2}$$

The opposite of -12 is 12.

$$c=\frac{12±6}{2}$$

Now solve the equation c=\frac{12±6}{2} when ± is plus. Add 12 to 6.

$$c=\frac{18}{2}$$

Divide 18 by 2.

$$c=9$$

Now solve the equation c=\frac{12±6}{2} when ± is minus. Subtract 6 from 12.

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

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 9 for x_{1} and 3 for x_{2}.

$$c^{2}-12c+27=\left(c-9\right)\left(c-3\right)$$

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.

$$x ^ 2 -12x +27 = 0$$

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

$$r + s = 12 rs = 27$$

Two numbers r and s sum up to 12 exactly when the average of the two numbers is \frac{1}{2}*12 = 6. 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>

$$r = 6 - u s = 6 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 27

$$(6 - u) (6 + u) = 27$$

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

$$36 - u^2 = 27$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = 27-36 = -9$$

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

$$u^2 = 9 u = \pm\sqrt{9} = \pm 3$$

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

$$r =6 - 3 = 3 s = 6 + 3 = 9$$

Evaluate

$$\left(c-9\right)\left(c-3\right)$$