Question

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

Answer 1

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)`