Solve 3w^2+14w+8

$$3 w ^ { 2 } + 14 w + 8$$

Factor

$$\left(w+4\right)\left(3w+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3w^{2}+aw+bw+8. To find a and b, set up a system to be solved.

$$a+b=14 ab=3\times 8=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 14.

$$a=2 b=12$$

Rewrite 3w^{2}+14w+8 as \left(3w^{2}+2w\right)+\left(12w+8\right).

$$\left(3w^{2}+2w\right)+\left(12w+8\right)$$

Factor out w in the first and 4 in the second group.

$$w\left(3w+2\right)+4\left(3w+2\right)$$

Factor out common term 3w+2 by using distributive property.

$$\left(3w+2\right)\left(w+4\right)$$

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.

$$3w^{2}+14w+8=0$$

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.

$$w=\frac{-14±\sqrt{14^{2}-4\times 3\times 8}}{2\times 3}$$

Square 14.

$$w=\frac{-14±\sqrt{196-4\times 3\times 8}}{2\times 3}$$

Multiply -4 times 3.

$$w=\frac{-14±\sqrt{196-12\times 8}}{2\times 3}$$

Multiply -12 times 8.

$$w=\frac{-14±\sqrt{196-96}}{2\times 3}$$

Add 196 to -96.

$$w=\frac{-14±\sqrt{100}}{2\times 3}$$

Take the square root of 100.

$$w=\frac{-14±10}{2\times 3}$$

Multiply 2 times 3.

$$w=\frac{-14±10}{6}$$

Now solve the equation w=\frac{-14±10}{6} when ± is plus. Add -14 to 10.

$$w=-\frac{4}{6}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

$$w=-\frac{2}{3}$$

Now solve the equation w=\frac{-14±10}{6} when ± is minus. Subtract 10 from -14.

$$w=-\frac{24}{6}$$

Divide -24 by 6.

$$w=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{3} for x_{1} and -4 for x_{2}.

$$3w^{2}+14w+8=3\left(w-\left(-\frac{2}{3}\right)\right)\left(w-\left(-4\right)\right)$$

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

$$3w^{2}+14w+8=3\left(w+\frac{2}{3}\right)\left(w+4\right)$$

Add \frac{2}{3} to w by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3w^{2}+14w+8=3\times \frac{3w+2}{3}\left(w+4\right)$$

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

$$3w^{2}+14w+8=\left(3w+2\right)\left(w+4\right)$$

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

$$x ^ 2 +\frac{14}{3}x +\frac{8}{3} = 0$$

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

$$r + s = -\frac{14}{3} rs = \frac{8}{3}$$

Two numbers r and s sum up to -\frac{14}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{14}{3} = -\frac{7}{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>

$$r = -\frac{7}{3} - u s = -\frac{7}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{3}

$$(-\frac{7}{3} - u) (-\frac{7}{3} + u) = \frac{8}{3}$$

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

$$\frac{49}{9} - u^2 = \frac{8}{3}$$

Simplify the expression by subtracting \frac{49}{9} on both sides

$$-u^2 = \frac{8}{3}-\frac{49}{9} = -\frac{25}{9}$$

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.

$$r =-\frac{7}{3} - \frac{5}{3} = -4 s = -\frac{7}{3} + \frac{5}{3} = -0.667$$

Evaluate

$$\left(w+4\right)\left(3w+2\right)$$