Factor the quadratic expression #20x^2 + 13x + 2# completely?

1 Answer
KillerBunny
Mar 27, 2015

A quadratic expression is completely factorizable if and only if its discriminant is positive. Given a quadratic expression of the form #ax^2+bx+c#, the discriminant #\Delta# is defined as #b^2-4ac#. In your case, we have #a=20#, #b=13# and #c=2#. For this values, we have #\Delta=9#, which means that we can factor the expression finding two solutions #x_1# and #x_2#, and thus writing #20x^2+13x+2=(x-x_1)(x-x_2)#.

To find the solutions, we have the formula
#x_{1,2}=\frac{-b\pm \sqrt(\Delta)}{2a}#.
Since #\Delta=9#, its square root equals 3. Plugging the values, we have the two solutions
#x_1={-13+3}/{40}=-1/4# and
#x_2={-13-3}/{40}=-2/5#.

The factorization is thus #(x+1/4)(x+2/5)#

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