Solve 6m^2-19m+3

$$6 m ^ { 2 } - 19 m + 3$$

Factor

$$\left(m-3\right)\left(6m-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6m^{2}+am+bm+3. To find a and b, set up a system to be solved.

$$a+b=-19 ab=6\times 3=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -19.

$$a=-18 b=-1$$

Rewrite 6m^{2}-19m+3 as \left(6m^{2}-18m\right)+\left(-m+3\right).

$$\left(6m^{2}-18m\right)+\left(-m+3\right)$$

Factor out 6m in the first and -1 in the second group.

$$6m\left(m-3\right)-\left(m-3\right)$$

Factor out common term m-3 by using distributive property.

$$\left(m-3\right)\left(6m-1\right)$$

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.

$$6m^{2}-19m+3=0$$

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.

$$m=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 6\times 3}}{2\times 6}$$

Square -19.

$$m=\frac{-\left(-19\right)±\sqrt{361-4\times 6\times 3}}{2\times 6}$$

Multiply -4 times 6.

$$m=\frac{-\left(-19\right)±\sqrt{361-24\times 3}}{2\times 6}$$

Multiply -24 times 3.

$$m=\frac{-\left(-19\right)±\sqrt{361-72}}{2\times 6}$$

Add 361 to -72.

$$m=\frac{-\left(-19\right)±\sqrt{289}}{2\times 6}$$

Take the square root of 289.

$$m=\frac{-\left(-19\right)±17}{2\times 6}$$

The opposite of -19 is 19.

$$m=\frac{19±17}{2\times 6}$$

Multiply 2 times 6.

$$m=\frac{19±17}{12}$$

Now solve the equation m=\frac{19±17}{12} when ± is plus. Add 19 to 17.

$$m=\frac{36}{12}$$

Divide 36 by 12.

$$m=3$$

Now solve the equation m=\frac{19±17}{12} when ± is minus. Subtract 17 from 19.

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

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$m=\frac{1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and \frac{1}{6} for x_{2}.

$$6m^{2}-19m+3=6\left(m-3\right)\left(m-\frac{1}{6}\right)$$

Subtract \frac{1}{6} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6m^{2}-19m+3=6\left(m-3\right)\times \frac{6m-1}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6m^{2}-19m+3=\left(m-3\right)\left(6m-1\right)$$

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 6

$$x ^ 2 -\frac{19}{6}x +\frac{1}{2} = 0$$

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

$$r + s = \frac{19}{6} rs = \frac{1}{2}$$

Two numbers r and s sum up to \frac{19}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{19}{6} = \frac{19}{12}. 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>

$$r = \frac{19}{12} - u s = \frac{19}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{2}

$$(\frac{19}{12} - u) (\frac{19}{12} + u) = \frac{1}{2}$$

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

$$\frac{361}{144} - u^2 = \frac{1}{2}$$

Simplify the expression by subtracting \frac{361}{144} on both sides

$$-u^2 = \frac{1}{2}-\frac{361}{144} = -\frac{289}{144}$$

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

$$u^2 = \frac{289}{144} u = \pm\sqrt{\frac{289}{144}} = \pm \frac{17}{12}$$

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

$$r =\frac{19}{12} - \frac{17}{12} = 0.167 s = \frac{19}{12} + \frac{17}{12} = 3$$

Evaluate

$$\left(m-3\right)\left(6m-1\right)$$