Solve 2y^2+5y-42

$$2 y ^ { 2 } + 5 y - 42$$

Factor

$$\left(2y-7\right)\left(y+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2y^{2}+ay+by-42. To find a and b, set up a system to be solved.

$$a+b=5 ab=2\left(-42\right)=-84$$

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 -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 5.

$$a=-7 b=12$$

Rewrite 2y^{2}+5y-42 as \left(2y^{2}-7y\right)+\left(12y-42\right).

$$\left(2y^{2}-7y\right)+\left(12y-42\right)$$

Factor out y in the first and 6 in the second group.

$$y\left(2y-7\right)+6\left(2y-7\right)$$

Factor out common term 2y-7 by using distributive property.

$$\left(2y-7\right)\left(y+6\right)$$

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.

$$2y^{2}+5y-42=0$$

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.

$$y=\frac{-5±\sqrt{5^{2}-4\times 2\left(-42\right)}}{2\times 2}$$

Square 5.

$$y=\frac{-5±\sqrt{25-4\times 2\left(-42\right)}}{2\times 2}$$

Multiply -4 times 2.

$$y=\frac{-5±\sqrt{25-8\left(-42\right)}}{2\times 2}$$

Multiply -8 times -42.

$$y=\frac{-5±\sqrt{25+336}}{2\times 2}$$

Add 25 to 336.

$$y=\frac{-5±\sqrt{361}}{2\times 2}$$

Take the square root of 361.

$$y=\frac{-5±19}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{-5±19}{4}$$

Now solve the equation y=\frac{-5±19}{4} when ± is plus. Add -5 to 19.

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

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

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

Now solve the equation y=\frac{-5±19}{4} when ± is minus. Subtract 19 from -5.

$$y=-\frac{24}{4}$$

Divide -24 by 4.

$$y=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{2} for x_{1} and -6 for x_{2}.

$$2y^{2}+5y-42=2\left(y-\frac{7}{2}\right)\left(y-\left(-6\right)\right)$$

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

$$2y^{2}+5y-42=2\left(y-\frac{7}{2}\right)\left(y+6\right)$$

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

$$2y^{2}+5y-42=2\times \frac{2y-7}{2}\left(y+6\right)$$

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

$$2y^{2}+5y-42=\left(2y-7\right)\left(y+6\right)$$

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

$$x ^ 2 +\frac{5}{2}x -21 = 0$$

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

$$r + s = -\frac{5}{2} rs = -21$$

Two numbers r and s sum up to -\frac{5}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{5}{2} = -\frac{5}{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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -21

$$(-\frac{5}{4} - u) (-\frac{5}{4} + u) = -21$$

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

$$\frac{25}{16} - u^2 = -21$$

Simplify the expression by subtracting \frac{25}{16} on both sides

$$-u^2 = -21-\frac{25}{16} = -\frac{361}{16}$$

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

$$u^2 = \frac{361}{16} u = \pm\sqrt{\frac{361}{16}} = \pm \frac{19}{4}$$

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

$$r =-\frac{5}{4} - \frac{19}{4} = -6 s = -\frac{5}{4} + \frac{19}{4} = 3.500$$

Evaluate

$$\left(2y-7\right)\left(y+6\right)$$