How to Solve Quadratic Equations & Graphs: ECZ Maths Tutorial
Quadratic equations ($ax^2 + bx + c = 0$) and parabolic graphs are heavily tested in Examination Council of Zambia (ECZ) Mathematics across Grade 9, Grade 12, and GCE papers. In Paper 2, candidates are guaranteed a 7-mark question requiring formula substitution or graph plotting.
This tutorial breaks down the Quadratic Formula Method, Completing the Square, and Graph Plotting Rules with worked examples.
1. Worked Example: Quadratic Formula Method
Worked Example 1
Question: Solve the equation $2x^2 - 7x + 3 = 0$, giving your answers correct to 2 decimal places.
Solution Step-by-Step:
- Identify coefficients: $a = 2$, $b = -7$, $c = 3$.
- Write formula: $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(3)}}{2(2)}$.
- Simplify inside square root: $b^2 - 4ac = 49 - 24 = 25$.
- Calculate roots: $x = \frac{7 \pm \sqrt{25}}{4} = \frac{7 \pm 5}{4}$.
- $x_1 = \frac{12}{4} = \mathbf{3.00}$.
- $x_2 = \frac{2}{4} = \mathbf{0.50}$.
2. Parabola Turning Point & Line of Symmetry
For a quadratic function $y = ax^2 + bx + c$:
- Line of Symmetry: Vertical line $x = -\frac{b}{2a}$.
- Turning Point: Substitute $x = -\frac{b}{2a}$ back into the equation to find $y$-coordinate.
Frequently Asked Questions (FAQs)
What is the quadratic formula used in ECZ Mathematics?
The quadratic formula is x = (-b +- sqrt(b^2 - 4ac)) / (2a). In ECZ Paper 2, candidates are required to give answers correct to 2 decimal places.
How do you find the turning point of a quadratic parabola y = ax^2 + bx + c?
The x-coordinate of the turning point is given by x = -b / (2a). Substitute this x-value back into the original quadratic equation to find the corresponding y-coordinate.
How do you determine if a parabola has a maximum or minimum turning point?
If a > 0 (positive x^2 coefficient), the curve opens upwards (U-shape) and has a MINIMUM turning point. If a < 0 (negative x^2 coefficient), the curve opens downwards (∩-shape) and has a MAXIMUM turning point.