Solve 3c^2-16c+5

$$3 c ^ { 2 } - 16 c + 5$$

Factor

$$\left(c-5\right)\left(3c-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3c^{2}+ac+bc+5. To find a and b, set up a system to be solved.

$$a+b=-16 ab=3\times 5=15$$

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 15.

$$-1,-15 -3,-5$$

Calculate the sum for each pair.

$$-1-15=-16 -3-5=-8$$

The solution is the pair that gives sum -16.

$$a=-15 b=-1$$

Rewrite 3c^{2}-16c+5 as \left(3c^{2}-15c\right)+\left(-c+5\right).

$$\left(3c^{2}-15c\right)+\left(-c+5\right)$$

Factor out 3c in the first and -1 in the second group.

$$3c\left(c-5\right)-\left(c-5\right)$$

Factor out common term c-5 by using distributive property.

$$\left(c-5\right)\left(3c-1\right)$$

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.

$$3c^{2}-16c+5=0$$

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.

$$c=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 3\times 5}}{2\times 3}$$

Square -16.

$$c=\frac{-\left(-16\right)±\sqrt{256-4\times 3\times 5}}{2\times 3}$$

Multiply -4 times 3.

$$c=\frac{-\left(-16\right)±\sqrt{256-12\times 5}}{2\times 3}$$

Multiply -12 times 5.

$$c=\frac{-\left(-16\right)±\sqrt{256-60}}{2\times 3}$$

Add 256 to -60.

$$c=\frac{-\left(-16\right)±\sqrt{196}}{2\times 3}$$

Take the square root of 196.

$$c=\frac{-\left(-16\right)±14}{2\times 3}$$

The opposite of -16 is 16.

$$c=\frac{16±14}{2\times 3}$$

Multiply 2 times 3.

$$c=\frac{16±14}{6}$$

Now solve the equation c=\frac{16±14}{6} when ± is plus. Add 16 to 14.

$$c=\frac{30}{6}$$

Divide 30 by 6.

$$c=5$$

Now solve the equation c=\frac{16±14}{6} when ± is minus. Subtract 14 from 16.

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

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$c=\frac{1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and \frac{1}{3} for x_{2}.

$$3c^{2}-16c+5=3\left(c-5\right)\left(c-\frac{1}{3}\right)$$

Subtract \frac{1}{3} from c by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3c^{2}-16c+5=3\left(c-5\right)\times \frac{3c-1}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3c^{2}-16c+5=\left(c-5\right)\left(3c-1\right)$$

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.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 -\frac{16}{3}x +\frac{5}{3} = 0$$

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

$$r + s = \frac{16}{3} rs = \frac{5}{3}$$

Two numbers r and s sum up to \frac{16}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{16}{3} = \frac{8}{3}. 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>

$$r = \frac{8}{3} - u s = \frac{8}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{3}

$$(\frac{8}{3} - u) (\frac{8}{3} + u) = \frac{5}{3}$$

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

$$\frac{64}{9} - u^2 = \frac{5}{3}$$

Simplify the expression by subtracting \frac{64}{9} on both sides

$$-u^2 = \frac{5}{3}-\frac{64}{9} = -\frac{49}{9}$$

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

$$u^2 = \frac{49}{9} u = \pm\sqrt{\frac{49}{9}} = \pm \frac{7}{3}$$

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

$$r =\frac{8}{3} - \frac{7}{3} = 0.333 s = \frac{8}{3} + \frac{7}{3} = 5$$

Evaluate

$$\left(c-5\right)\left(3c-1\right)$$