Quadratic equations and inequalities are the algebraic engine of A-Math. Nature-of-roots, line–curve intersection and “find the range of values of k” questions recur in both Paper 1 and Paper 2, and the same discriminant reasoning feeds directly into later topics such as circles, curve sketching and calculus.
These notes cover the discriminant b² − 4ac and what its sign says about the roots and the graph, solving quadratic inequalities with a four-step method, and solving one linear and one non-linear equation by substitution, which links straight back to the discriminant through line–curve intersection.
What these notes cover
- •The discriminant and nature of roots
- •Nature of roots and the graph
- •Always positive and always negative
- •Proving roots are real and distinct for all k
- •Solving quadratic inequalities: the four-step method
- •Showing the solution on a number line
- •The sign test
- •Finding coefficients from a solution set
- •Simultaneous equations by substitution
- •Line–curve intersection
- •Common mistakes and exam technique
- •Summary
Key definitions
- Discriminant
- The expression D = b² − 4ac for the quadratic equation ax² + bx + c = 0, where a ≠ 0. Its sign decides the nature of the roots.
- Two real equal (repeated) roots
- When D = 0, √D = 0, so both signs in the quadratic formula give the same root, x = −b ÷ (2a). The curve touches the x-axis.
- Critical values
- The roots α and β (α < β) of the quadratic set equal to 0. They are only the boundaries of the region in an inequality answer, not the answer itself.
- Non-linear equation
- An equation that contains a term such as x², y² or xy. In A-Math it is usually paired with one linear equation and solved by substitution.
- Tangent
- A line that touches the curve at exactly one point. The substituted quadratic has two equal (repeated) real roots, so b² − 4ac = 0.
Key formulas
Quadratic formula
Gives the roots of ax² + bx + c = 0, where a ≠ 0. Rearrange into this form before reading off a, b and c.
Discriminant and nature of roots
D = b² − 4ac
D > 0: two real distinct roots (curve cuts the x-axis twice). D = 0: two real equal roots (touches the x-axis). D < 0: no real roots (does not meet the x-axis). D ≥ 0: real roots.
Always positive
ax² + bx + c > 0 for all real x ⇔ a > 0 and b² − 4ac < 0
Always negative needs a < 0 and b² − 4ac < 0. Both conditions are needed. For non-negative (≥ 0 for all x), use a > 0 and b² − 4ac ≤ 0.
Less than zero: between the roots
(x − α)(x − β) < 0 ⇔ α < x < β
For critical values α < β. The non-strict form (x − α)(x − β) ≤ 0 gives α ≤ x ≤ β. Also holds for a(x − α)(x − β) whenever a > 0.
Greater than zero: outside the roots
(x − α)(x − β) > 0 ⇔ x < α or x > β
For critical values α < β. The non-strict form gives x ≤ α or x ≥ β. The answer is always two separate pieces joined by “or”.
Line–curve intersection
b² − 4ac > 0: cuts at two distinct points; b² − 4ac = 0: tangent; b² − 4ac < 0: does not meet; b² − 4ac ≥ 0: meets
Apply to the quadratic formed after substituting the line into the curve, never to the curve's own equation. “Meets” or “intersects” without “distinct” means b² − 4ac ≥ 0.
Worked example
The line y = kx − 2 and the curve y = x² + 2x + 2 are given, where k is a constant. Find the range of values of k for which the line intersects the curve at two distinct points.
- 1Substitute the line into the curve by equating the two expressions for y: x² + 2x + 2 = kx − 2.
- 2Collect into the form ax² + bx + c = 0, grouping both x terms in one bracket: x² + (2 − k)x + 4 = 0, so a = 1, b = 2 − k and c = 4.
- 3Find the discriminant: D = (2 − k)² − 4(1)(4) = 4 − 4k + k² − 16 = k² − 4k − 12.
- 4Two distinct intersection points means two distinct real roots, so D > 0: k² − 4k − 12 > 0, which factorises to (k − 6)(k + 2) > 0.
- 5The graph of D against k is U-shaped and cuts the k-axis at −2 and 6, so D > 0 is the part above the axis: the two outer regions.
- 6Hence k < −2 or k > 6.
Watch out: Use the discriminant of the substituted quadratic, not of the curve's own equation, and never squash the answer into −2 > k > 6, because no number is below −2 and above 6 at once.
Common mistakes
Real roots
Wrong: x² + 4x + k = 0 has real roots, so 16 − 4k > 0, giving k < 4.
Right: Real includes equal roots: 16 − 4k ≥ 0, so k ≤ 4.
Always positive
Wrong: kx² + 4x + k > 0 for all x: use only 16 − 4k² < 0, giving k < −2 or k > 2.
Right: Need both k > 0 (curve opens upward) and 16 − 4k² < 0, so k > 2 only. “Always negative” needs a < 0 instead.
Taking square roots
Wrong: x² > 9 ⇒ x > ±3
Right: (x − 3)(x + 3) > 0, so x < −3 or x > 3. Similarly, x² < 9 gives −3 < x < 3.
Questions students ask
How do you find the nature of roots of a quadratic equation?
Rearrange the equation into ax² + bx + c = 0 and work out the discriminant b² − 4ac. If it is positive there are two real distinct roots, if it is zero there are two real equal (repeated) roots, and if it is negative there are no real roots.
Is the condition for real roots b² − 4ac > 0 or b² − 4ac ≥ 0?
Use b² − 4ac ≥ 0. “Real roots” includes the equal-roots case, so using > drops the end-points of the final range and costs the accuracy mark.
How do you solve a quadratic inequality in A-Math?
Rearrange so one side is 0 and the x² coefficient is positive, find the critical values α and β, sketch a U-shaped parabola through them, then read off the region. For < 0 the answer lies between the roots, α < x < β; for > 0 it lies outside, x < α or x > β.
How do you find the range of k for a line to meet or be a tangent to a curve?
Substitute the line into the curve to get one quadratic in x and rearrange it into ax² + bx + c = 0. Then apply the discriminant of that quadratic: b² − 4ac = 0 for a tangent, > 0 for two distinct points, ≥ 0 for meets or intersects, and < 0 for does not meet.
In the full PDF
- •Graphs showing how the sign of D appears on the curve, and the curve y = x² cut by three parallel lines
- •Number-line conventions with open and filled circles, plus the sign-test alternative to sketching
- •Further worked examples with M1/A1 mark tags, including always-positive ranges, “for all real k” proofs and rebuilding a quadratic from its solution set
- •Full simultaneous-equation solutions and tangent points of contact with the perfect-square check
- •The remaining common mistakes and the keyword → discriminant exam-technique table
- •A one-page summary of the whole topic