SL 2.7—Quadratic equations and inequalities

Solve quadratic equations and inequalities with sign control.

Use factorization, completing the square or the formula for equations; for inequalities, test intervals or use the parabola’s sign.

Worked example
(x−2)(x+1)≥0 gives x≤−1 or x≥2.

Worked example
Why are there two intervals? the product is positive outside the roots for an upward parabola.

Common boundary
Solving the equation alone does not solve the inequality.

Quadratic formula and discriminant: for $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and $\Delta=b^2-4ac$. If $\Delta>0$ there are two distinct real roots, $\Delta=0$ gives one repeated real root, and $\Delta<0$ gives no real roots. For $2x^2-3x-2=0$, $\Delta=25$, so $x=(3\pm5)/4$, giving $x=2$ or $x=-1/2$.