Solving Quadratic Inequalities

From roots to ranges

A quadratic equation asks for values that make an expression equal zero. A quadratic inequality asks where that expression is positive or negative, so its answer is usually a range of values. For example, to solve $x^2-x-2>0$, first find where $x^2-x-2=0$. Factorising gives $(x-2)(x+1)=0$, so the roots are $-1$ and $2$.

The graph of $y=x^2-x-2$ is U-shaped because the coefficient of $x^2$ is positive. It crosses the $x$-axis at the roots. Between them the curve is below the axis, so the expression is negative; outside them it is above the axis, so the expression is positive. Hence when . The word ‘or’ matters: these are two separate parts of the .

Data for Where a quadratic is positive or negative

Rearrange, solve, then choose the intervals

Consider $3x^2+2x-6>x^2+4x-2$. Subtract the right-hand side to obtain , then divide by to obtain . Its are and , and the U-shaped sketch shows that the required values lie outside them: or . By contrast, would be satisfied the roots, giving .

The roots themselves make the expression zero. They are therefore excluded by $<$ and $>$ but included by $\leq$ and $\geq$. For instance, $x^2-x-2\leq0$ gives .

You can make a negative coefficient of $x^2$ positive before sketching, but multiplying or dividing an inequality by a negative number reverses its sign. Avoid dividing by an expression containing $x$: its sign may change with $x$, and it may be zero. If a quadratic does not cross the $x$-axis twice, do not force an ‘inside or outside two roots’ answer; use its sketch to decide where, if anywhere, the required sign occurs.