Solve 3m^2+8m-3

$$3 m ^ { 2 } + 8 m - 3$$

Factor

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

Solution Steps

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

$$a+b=8 ab=3\left(-3\right)=-9$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -9.

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

Calculate the sum for each pair.

$$-1+9=8 -3+3=0$$

The solution is the pair that gives sum 8.

$$a=-1 b=9$$

Rewrite 3m^{2}+8m-3 as \left(3m^{2}-m\right)+\left(9m-3\right).

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

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

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

Factor out common term 3m-1 by using distributive property.

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

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.

$$3m^{2}+8m-3=0$$

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.

$$m=\frac{-8±\sqrt{8^{2}-4\times 3\left(-3\right)}}{2\times 3}$$

Square 8.

$$m=\frac{-8±\sqrt{64-4\times 3\left(-3\right)}}{2\times 3}$$

Multiply -4 times 3.

$$m=\frac{-8±\sqrt{64-12\left(-3\right)}}{2\times 3}$$

Multiply -12 times -3.

$$m=\frac{-8±\sqrt{64+36}}{2\times 3}$$

Add 64 to 36.

$$m=\frac{-8±\sqrt{100}}{2\times 3}$$

Take the square root of 100.

$$m=\frac{-8±10}{2\times 3}$$

Multiply 2 times 3.

$$m=\frac{-8±10}{6}$$

Now solve the equation m=\frac{-8±10}{6} when ± is plus. Add -8 to 10.

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

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

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

Now solve the equation m=\frac{-8±10}{6} when ± is minus. Subtract 10 from -8.

$$m=-\frac{18}{6}$$

Divide -18 by 6.

$$m=-3$$

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

$$3m^{2}+8m-3=3\left(m-\frac{1}{3}\right)\left(m-\left(-3\right)\right)$$

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

$$3m^{2}+8m-3=3\left(m-\frac{1}{3}\right)\left(m+3\right)$$

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

$$3m^{2}+8m-3=3\times \frac{3m-1}{3}\left(m+3\right)$$

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

$$3m^{2}+8m-3=\left(3m-1\right)\left(m+3\right)$$

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

$$x ^ 2 +\frac{8}{3}x -1 = 0$$

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

$$r + s = -\frac{8}{3} rs = -1$$

Two numbers r and s sum up to -\frac{8}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{3} = -\frac{4}{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>

$$r = -\frac{4}{3} - u s = -\frac{4}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(-\frac{4}{3} - u) (-\frac{4}{3} + u) = -1$$

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

$$\frac{16}{9} - u^2 = -1$$

Simplify the expression by subtracting \frac{16}{9} on both sides

$$-u^2 = -1-\frac{16}{9} = -\frac{25}{9}$$

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

$$u^2 = \frac{25}{9} u = \pm\sqrt{\frac{25}{9}} = \pm \frac{5}{3}$$

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

$$r =-\frac{4}{3} - \frac{5}{3} = -3 s = -\frac{4}{3} + \frac{5}{3} = 0.333$$

Evaluate

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