Solve 14y^2+37y+24

$$14 { y }^{ 2 } +37y+24$$

Factor

$$\left(2y+3\right)\left(7y+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 14y^{2}+ay+by+24. To find a and b, set up a system to be solved.

$$a+b=37 ab=14\times 24=336$$

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 336.

$$1,336 2,168 3,112 4,84 6,56 7,48 8,42 12,28 14,24 16,21$$

Calculate the sum for each pair.

$$1+336=337 2+168=170 3+112=115 4+84=88 6+56=62 7+48=55 8+42=50 12+28=40 14+24=38 16+21=37$$

The solution is the pair that gives sum 37.

$$a=16 b=21$$

Rewrite 14y^{2}+37y+24 as \left(14y^{2}+16y\right)+\left(21y+24\right).

$$\left(14y^{2}+16y\right)+\left(21y+24\right)$$

Factor out 2y in the first and 3 in the second group.

$$2y\left(7y+8\right)+3\left(7y+8\right)$$

Factor out common term 7y+8 by using distributive property.

$$\left(7y+8\right)\left(2y+3\right)$$

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.

$$14y^{2}+37y+24=0$$

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.

$$y=\frac{-37±\sqrt{37^{2}-4\times 14\times 24}}{2\times 14}$$

Square 37.

$$y=\frac{-37±\sqrt{1369-4\times 14\times 24}}{2\times 14}$$

Multiply -4 times 14.

$$y=\frac{-37±\sqrt{1369-56\times 24}}{2\times 14}$$

Multiply -56 times 24.

$$y=\frac{-37±\sqrt{1369-1344}}{2\times 14}$$

Add 1369 to -1344.

$$y=\frac{-37±\sqrt{25}}{2\times 14}$$

Take the square root of 25.

$$y=\frac{-37±5}{2\times 14}$$

Multiply 2 times 14.

$$y=\frac{-37±5}{28}$$

Now solve the equation y=\frac{-37±5}{28} when ± is plus. Add -37 to 5.

$$y=-\frac{32}{28}$$

Reduce the fraction \frac{-32}{28} to lowest terms by extracting and canceling out 4.

$$y=-\frac{8}{7}$$

Now solve the equation y=\frac{-37±5}{28} when ± is minus. Subtract 5 from -37.

$$y=-\frac{42}{28}$$

Reduce the fraction \frac{-42}{28} to lowest terms by extracting and canceling out 14.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{8}{7} for x_{1} and -\frac{3}{2} for x_{2}.

$$14y^{2}+37y+24=14\left(y-\left(-\frac{8}{7}\right)\right)\left(y-\left(-\frac{3}{2}\right)\right)$$

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

$$14y^{2}+37y+24=14\left(y+\frac{8}{7}\right)\left(y+\frac{3}{2}\right)$$

Add \frac{8}{7} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$14y^{2}+37y+24=14\times \frac{7y+8}{7}\left(y+\frac{3}{2}\right)$$

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

$$14y^{2}+37y+24=14\times \frac{7y+8}{7}\times \frac{2y+3}{2}$$

Multiply \frac{7y+8}{7} times \frac{2y+3}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$14y^{2}+37y+24=14\times \frac{\left(7y+8\right)\left(2y+3\right)}{7\times 2}$$

Multiply 7 times 2.

$$14y^{2}+37y+24=14\times \frac{\left(7y+8\right)\left(2y+3\right)}{14}$$

Cancel out 14, the greatest common factor in 14 and 14.

$$14y^{2}+37y+24=\left(7y+8\right)\left(2y+3\right)$$

Evaluate

$$\left(2y+3\right)\left(7y+8\right)$$