Solve 2y^2+13y+21

$$2 y ^ { 2 } + 13 y + 21$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2y^{2}+ay+by+21. To find a and b, set up a system to be solved.

$$a+b=13 ab=2\times 21=42$$

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 42.

$$1,42 2,21 3,14 6,7$$

Calculate the sum for each pair.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

The solution is the pair that gives sum 13.

$$a=6 b=7$$

Rewrite 2y^{2}+13y+21 as \left(2y^{2}+6y\right)+\left(7y+21\right).

$$\left(2y^{2}+6y\right)+\left(7y+21\right)$$

Factor out 2y in the first and 7 in the second group.

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

Factor out common term y+3 by using distributive property.

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

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.

$$2y^{2}+13y+21=0$$

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.

$$y=\frac{-13±\sqrt{13^{2}-4\times 2\times 21}}{2\times 2}$$

Square 13.

$$y=\frac{-13±\sqrt{169-4\times 2\times 21}}{2\times 2}$$

Multiply -4 times 2.

$$y=\frac{-13±\sqrt{169-8\times 21}}{2\times 2}$$

Multiply -8 times 21.

$$y=\frac{-13±\sqrt{169-168}}{2\times 2}$$

Add 169 to -168.

$$y=\frac{-13±\sqrt{1}}{2\times 2}$$

Take the square root of 1.

$$y=\frac{-13±1}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{-13±1}{4}$$

Now solve the equation y=\frac{-13±1}{4} when ± is plus. Add -13 to 1.

$$y=-\frac{12}{4}$$

Divide -12 by 4.

$$y=-3$$

Now solve the equation y=\frac{-13±1}{4} when ± is minus. Subtract 1 from -13.

$$y=-\frac{14}{4}$$

Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.

$$y=-\frac{7}{2}$$

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{7}{2} for x_{2}.

$$2y^{2}+13y+21=2\left(y-\left(-3\right)\right)\left(y-\left(-\frac{7}{2}\right)\right)$$

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

$$2y^{2}+13y+21=2\left(y+3\right)\left(y+\frac{7}{2}\right)$$

Add \frac{7}{2} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2y^{2}+13y+21=2\left(y+3\right)\times \frac{2y+7}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2y^{2}+13y+21=\left(y+3\right)\left(2y+7\right)$$

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 2

$$x ^ 2 +\frac{13}{2}x +\frac{21}{2} = 0$$

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

$$r + s = -\frac{13}{2} rs = \frac{21}{2}$$

Two numbers r and s sum up to -\frac{13}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{2} = -\frac{13}{4}. 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>

$$r = -\frac{13}{4} - u s = -\frac{13}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{21}{2}

$$(-\frac{13}{4} - u) (-\frac{13}{4} + u) = \frac{21}{2}$$

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

$$\frac{169}{16} - u^2 = \frac{21}{2}$$

Simplify the expression by subtracting \frac{169}{16} on both sides

$$-u^2 = \frac{21}{2}-\frac{169}{16} = -\frac{1}{16}$$

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

$$u^2 = \frac{1}{16} u = \pm\sqrt{\frac{1}{16}} = \pm \frac{1}{4}$$

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

$$r =-\frac{13}{4} - \frac{1}{4} = -3.500 s = -\frac{13}{4} + \frac{1}{4} = -3$$

Evaluate

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