Quadratic Functions and Equations in One Variable · 1.1.5
Roots of a quadratic equation
The roots of a quadratic equation ax² + bx + c = 0 are the values of x that satisfy the equation, and they are also the x-intercepts of the graph of f(x) = ax² + bx + c. This standard focuses on understanding what roots represent, before students learn to calculate them.
The official learning standard (1.1.5)
“Explain the meaning of roots of a quadratic equation. Exploratory activities need to be carried out.
Limited to real roots. The position of the roots on the graphs of quadratic equations needs to be discussed.”
What it means
The roots of a quadratic equation ax² + bx + c = 0 are the values of x that satisfy the equation, and they are also the x-intercepts of the graph of f(x) = ax² + bx + c. This standard focuses on understanding what roots represent, before students learn to calculate them.
How it is examined
Paper 1 tests this conceptually, for example asking students to state the roots directly from a given graph or to explain what a root represents. It also underlies Paper 2 questions that combine solving equations with sketching or interpreting graphs.
Worked example
The graph of a quadratic function f(x) = x² − x − 6 crosses the x-axis at x = −2 and x = 3. State what these two x-values represent, and verify that they satisfy f(x) = 0.
- The points where the graph crosses the x-axis are the values of x for which f(x) = 0.
- These x-values are called the roots of the quadratic equation x² − x − 6 = 0.
- Check x = −2: f(−2) = (−2)² − (−2) − 6 = 4 + 2 − 6 = 0. ✓
- Check x = 3: f(3) = (3)² − 3 − 6 = 9 − 3 − 6 = 0. ✓
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What exactly is a "root" of a quadratic equation?
A root is a value of x that makes the equation ax² + bx + c = 0 true. Graphically, it is the x-coordinate of a point where the curve f(x) = ax² + bx + c meets the x-axis, since f(x) = 0 there.
Can a quadratic equation have no real roots?
Yes. If the graph of f(x) = ax² + bx + c never touches the x-axis, for example a parabola that opens upward entirely above the x-axis, then there is no real value of x for which f(x) = 0, so the equation has no real roots.
How many roots can a quadratic equation have?
A quadratic equation can have at most two real roots. They can be two distinct real roots, one repeated (equal) root where the graph just touches the x-axis, or no real roots at all when the graph does not meet the x-axis.