Quadratic Functions and Equations
This topic covers quadratic functions, expressed as y = ax2 + bx + c. Mastering the links between their algebraic forms and graphical properties (parabolas) is essential for solving equations and inequalities efficiently without a calculator.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The graph of y = ax2 + bx + c is a parabola. If a > 0, it is a 'U' shape with a minimum point. If a < 0, it is an 'n' shape with a maximum point.
- Completing the square transforms the quadratic into a(x - p)2 + q, which directly reveals the turning point (vertex) of the parabola at coordinates (p, q).
- The discriminant, Δ = b2 - 4ac, determines the number of real roots (x-intercepts): Δ > 0 gives two distinct roots, Δ = 0 gives one repeated root (the vertex is on the x-axis), and Δ < 0 gives no real roots.
- The solutions to the equation ax2 + bx + c = 0 are called the roots. These can be found by factorising, completing the square, or using the quadratic formula.
- A quadratic graph is symmetrical. The line of symmetry is a vertical line passing through the vertex, with the equation x = -b/(2a).
Diagram
Formulae
x = [-b ± √(b2 - 4ac)] / (2a) To find the roots of any quadratic equation ax2 + bx + c = 0, especially when it cannot be easily factorised.
a(x + p)2 + q, where p = b/(2a) and q = c - b2/(4a) This is the 'completed square' form. Use it to find the coordinates of the minimum or maximum point, (-p, q), and to solve equations.
Definitions
- Quadratic Function
- A function that can be written in the form f(x) = ax2 + bx + c, where a, b, and c are constants and a is not equal to zero.
- Discriminant
- The expression b2 - 4ac from within the quadratic formula. Its sign reveals the nature of the roots of a quadratic equation.
- Root
- A solution to a quadratic equation ax2 + bx + c = 0. On a graph, the real roots are the x-coordinates where the parabola intersects the x-axis.
Worked example
The quadratic equation 2x2 + (k+1)x + 2 = 0 has exactly one real root. Find the possible values of k.
- 1
For exactly one real root, the discriminant must be zero:
b2 - 4ac = 0 - 2
Identify the coefficients from the equation:
a = 2, b = (k+1), c = 2 - 3
Substitute these values into the discriminant equation:
(k+1)2 - 4(2)(2) = 0.
- 4
Simplify the equation:
(k+1)2 - 16 = 0.
- 5
This is a difference of two squares:
[(k+1) - 4][(k+1) + 4] = 0.
- 6
Simplify the factors:
(k-3)(k+5) = 0 - 7
Solve for k:
k - 3 = 0 or k + 5 = 0 - 8
The possible values for k are 3 and -5.
Answer: k = 3 or k = -5
Common mistakes
- ×Sign errors when using the quadratic formula, especially with the -b and -4ac terms when b or c are negative. For b=-5, -b is +5.
- ×Forgetting constraints on coefficients. If an equation is given as kx2 + ⋯, it is only a quadratic if k is not zero. This can affect the valid range of solutions in inequality problems.
- ×Mistakes when completing the square for ax2⋯ where a is not 1. You must factor out 'a' from the first two terms first: a(x2 + (b/a)x) + c.
- ×Incorrectly solving quadratic inequalities. After finding the critical values (the roots), always sketch the parabola to determine whether the solution is between the roots or outside the roots.
No-calculator tips
- ✓Always try to factorise before using the formula. It's much faster and less prone to arithmetic errors.
- ✓Before solving, quickly check the discriminant. If b2 - 4ac is a perfect square, the quadratic will factorise. If it's negative, you can state 'no real solutions' immediately.
- ✓If all coefficients in an equation are multiples of a number, divide the entire equation by that number to work with smaller, simpler values.
- ✓The x-coordinate of the vertex, x = -b/(2a), is the average of the two roots. This can be a quick way to find the line of symmetry or vertex if you know the roots.