Concept 09 / 10Complex Numbers and Quadratic Equations
Sign of a Quadratic Expression and Location of Roots
When ax^2 + bx + c keeps one sign for all real x, the range of a quadratic via its vertex, solving quadratic inequalities, and the three conditions (discriminant, f(k), vertex) for placing roots relative to a number or inside an interval.
The quadratic can be understood completely from its graph, a parabola with vertex at . Completing the square gives
.
So the vertex is . If the parabola opens upward, the minimum value is and the range is . If the maximum value is .
Sign for all real . From the completed square:
for all real and .
for all real and .
for all real and .
If the leading coefficient contains a parameter, check the case separately, since then the expression is linear and can only keep one sign if it is constant.
Sign between and outside the roots. If with real roots , then has the sign of outside and the opposite sign strictly between them. This solves every quadratic inequality: for , and or . For rational inequalities , use the wavy-curve (sign chart) method: mark the zeros of the numerator and denominator on the number line, and note that the sign changes at each simple zero and does not change at a zero of even multiplicity. Never multiply both sides by without knowing its sign.
Location of roots. Many questions ask for the parameter values that place the roots in a certain position relative to a number or an interval. Three quantities decide everything, all read off the graph: the discriminant (whether the roots are real), the value (which side of the parabola's zeros is on) and the vertex (which side of the axis of symmetry is on). Take ; for replace by throughout.
Both roots greater than : , and .
Both roots less than : , and .
strictly between the roots: alone. The figure shows why: an upward parabola that dips below the axis at must cross the axis on both sides of , so real roots follow automatically and no discriminant condition is needed.
Both roots strictly inside : , , and .
Exactly one root in : (then check the endpoints separately if the question allows a root there).
and both between the roots: and .
Why is the vertex condition needed in the 'both greater than ' case? Because only says that is outside the interval between the roots; it could be to the left of both or to the right of both. The position of the axis of symmetry tells us which.
Key Takeaways:
- ; the extreme value is at .
- for all ; for all .
- Between the roots has the sign opposite to ; outside them, the same sign as .
- between the roots .
- Both roots greater than : , , .
- Exactly one root in : .
- In rational inequalities, never cross-multiply by an expression of unknown sign.
- Pitfall: forgetting the case where the leading coefficient containing a parameter becomes .
Brain-Teaser 1: Find all for which for all real .
Solution: Case : the expression is the constant , which works. Case : we need and . , which is negative for . Together with this gives . Including , the answer is . Many students lose by applying the quadratic condition without checking the degenerate case.
Brain-Teaser 2: Find such that both roots of are less than .
Solution: The leading coefficient is . Condition 1: , where , so . Condition 2: , where , so or . Condition 3: the vertex is less than , so . Intersecting all three gives . Check with : the equation is with roots and , and is not less than , so is correctly excluded.
Brain-Teaser 3: For what does exactly one root of lie in ?
Solution: Let . Then and . Since , for exactly one root in we need , so . (If , both and are negative, and since the parabola opens upward, both roots are outside .) Also check the boundary : then , so is a root, which is not in the open interval; the other root is , so there is no root in . The answer is . Notice that tells us immediately that lies between the roots, so exactly one root is greater than and the question is only about the other.
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