Solve 2y^2-13y-7

$$2 { y }^{ 2 } -13y-7$$

Factor

$$\left(y-7\right)\left(2y+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2y^{2}+ay+by-7. To find a and b, set up a system to be solved.

$$a+b=-13 ab=2\left(-7\right)=-14$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -13.

$$a=-14 b=1$$

Rewrite 2y^{2}-13y-7 as \left(2y^{2}-14y\right)+\left(y-7\right).

$$\left(2y^{2}-14y\right)+\left(y-7\right)$$

Factor out 2y in 2y^{2}-14y.

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

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

$$\left(y-7\right)\left(2y+1\right)$$

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.

$$2y^{2}-13y-7=0$$

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.

$$y=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Square -13.

$$y=\frac{-\left(-13\right)±\sqrt{169-4\times 2\left(-7\right)}}{2\times 2}$$

Multiply -4 times 2.

$$y=\frac{-\left(-13\right)±\sqrt{169-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$y=\frac{-\left(-13\right)±\sqrt{169+56}}{2\times 2}$$

Add 169 to 56.

$$y=\frac{-\left(-13\right)±\sqrt{225}}{2\times 2}$$

Take the square root of 225.

$$y=\frac{-\left(-13\right)±15}{2\times 2}$$

The opposite of -13 is 13.

$$y=\frac{13±15}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{13±15}{4}$$

Now solve the equation y=\frac{13±15}{4} when ± is plus. Add 13 to 15.

$$y=\frac{28}{4}$$

Divide 28 by 4.

$$y=7$$

Now solve the equation y=\frac{13±15}{4} when ± is minus. Subtract 15 from 13.

$$y=-\frac{2}{4}$$

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

$$y=-\frac{1}{2}$$

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

$$2y^{2}-13y-7=2\left(y-7\right)\left(y-\left(-\frac{1}{2}\right)\right)$$

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

$$2y^{2}-13y-7=2\left(y-7\right)\left(y+\frac{1}{2}\right)$$

Add \frac{1}{2} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2y^{2}-13y-7=2\left(y-7\right)\times \frac{2y+1}{2}$$

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

$$2y^{2}-13y-7=\left(y-7\right)\left(2y+1\right)$$

Evaluate

$$\left(y-7\right)\left(2y+1\right)$$