Quadratic Equations & The Quadratic Formula
A quadratic expression takes the form ax² + bx + c (where a ≠ 0) and plots as a symmetrical parabola. In GCSE Maths, you solve quadratic equations by factorising into two brackets, using the quadratic formula, or completing the square to find the roots, y-intercept, and turning point.
- Quadratic Equation
- An equation of the second degree, meaning it contains at least one term that is squared.
- Factorising
- Writing an expression as a product of its factors, typically putting it into brackets.
- Quadratic Formula
- x = (-b ± √(b² - 4ac)) / (2a)
Quadratic Parabola Plotter, Roots, Turning Point & Quadratic Formula
Adjust coefficients a, b, and c in y = ax² + bx + c to see how the parabola, roots (x-intercepts), and turning point move on the graph.
y = x² - 2x - 3
(-2)² − 4(1)(-3) = 16 → 2 real roots (crosses x-axis twice)x = 1.00b in brackets when squaring: (-2)² = 4 (always positive!). Typing -2² into a calculator without brackets gives -4 and loses the accuracy mark.Exam notes: All 4 Ways to Solve a Quadratic Equation + Turning Points
- Every quadratic equation must be rearranged into standard form ax² + bx + c = 0 (equal to zero) before you try to factorise or apply the quadratic formula. Never read off coefficients a, b, c while numbers are still on the right-hand side.
- The Quadratic Formula is x = [−b ± √(b² − 4ac)] ÷ (2a). When b is negative (e.g. b = −6), always put negative b in brackets when squaring: (−6)² = +36, never −36.
- The Discriminant Δ = b² − 4ac tells you the number of real roots: if b² − 4ac > 0, there are 2 distinct real roots (parabola crosses the x-axis twice); if b² − 4ac = 0, there is 1 repeated root (vertex touches the x-axis); if b² − 4ac < 0, there are 0 real roots.
- Completing the square rewrites x² + bx + c into a(x + p)² + q. The turning point (minimum if a > 0, maximum if a < 0) is at coordinates (−p, q), and the line of symmetry is x = −p.
| Method | When to Use It | Exact Procedure / Formula |
|---|---|---|
| 1. Factorising into (x + m)(x + n) = 0 | Non-calculator paper or integer roots | Find two numbers that multiply to ac and add to b; set each bracket = 0 |
| 2. Difference of Two Squares (a² − b²) | Two-term quadratic with no x term (e.g. x² − 49 = 0) | Factorises immediately into (x − 7)(x + 7) = 0 ⇒ x = 7 or x = −7 |
| 3. Quadratic Formula | Calculator paper or asks for '2 decimal places / 3 sig figs' | x = [−b ± √(b² − 4ac)] / (2a) |
| 4. Completing the Square a(x + p)² + q | Question asks for turning point coordinates or exact surd form | Halve the coefficient of x: (x + b/2)² − (b/2)² + c; turning point is (−p, q) |
Key Concepts & Exam Definitions (8)
- Quadratic Equation
- An equation of the second degree, meaning it contains at least one term that is squared.
- Factorising
- Writing an expression as a product of its factors, typically putting it into brackets.
- Quadratic Formula
- x = (-b ± √(b² - 4ac)) / (2a)
- Completing the Square
- A method of writing a quadratic expression ax² + bx + c in the form a(x + p)² + q.
- Parabola
- The distinctive U-shaped curve that is the graph of a quadratic function.
- The Quadratic Formula & Discriminant (b² − 4ac)
- For ax² + bx + c = 0 (with a ≠ 0), x = [−b ± √(b² − 4ac)] / (2a). The expression under the square root, b² − 4ac, determines whether the quadratic has two, one, or zero real solutions.
- Turning Point from Completed Square Form
- When y = a(x − h)² + k, the minimum (or maximum) value of y occurs when the squared bracket (x − h)² = 0, giving the turning point directly as (h, k).
- Quadratic Inequalities (Critical Values Method)
- 1) Solve ax² + bx + c = 0 to find the two critical values x₁ and x₂. 2) Sketch a U-shaped parabola: if ax² + bx + c < 0 (below the axis), the solution is the single inside region x₁ < x < x₂; if ax² + bx + c > 0 (above the axis), the solution is the two outside regions x < x₁ or x > x₂.
Quick-Fire Flashcards (8)
Click any card to check the answer, then mark it as Known to track your progress.
What is the quadratic formula?
What shape is the graph of y = x²?
Factorise x² + 5x + 6
In the expression (x+a)², what is the expanded constant term?
What is the discriminant?
State the Quadratic Formula for solving ax² + bx + c = 0.
What is the turning point and line of symmetry of y = (x − 4)² − 9?
How do you spot in a GCSE exam that you must use the Quadratic Formula instead of factorising?
Exam-Style Questions & Mark Schemes (5)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
Solve x² - 9 = 0
Click “Check Mark Scheme” to compare your answer, x = 3 or x = -3 (Difference of two squares)
- Q2
Factorise and solve x² - 7x + 10 = 0
Click “Check Mark Scheme” to compare your answer, (x - 2)(x - 5) = 0, so x = 2 or x = 5
- Q3
Find the turning point of y = (x - 3)² + 4
Click “Check Mark Scheme” to compare your answer, (3, 4)
- Q4
Solve 2x² − 7x − 3 = 0, giving your answers correct to 2 decimal places. [3 marks]
Click “Check Mark Scheme” to compare your answer, 1) Identify a = 2, b = −7, c = −3 and substitute into the quadratic formula: x = [ −(−7) ± √((−7)² − 4(2)(−3)) ] ÷ (2 × 2) [1 mark]. 2) Simplify inside the root: x = [ 7 ± √(49 + 24) ] ÷ 4 = (7 ± √73) ÷ 4 [1 mark]. 3) Evaluate both roots: x = (7 + 8.544) ÷ 4 = 3.89 or x = (7 − 8.544) ÷ 4 = −0.39 (to 2 d.p) [1 mark].
- Q5
Write x² − 8x + 5 in the form (x − a)² + b, and hence state the coordinates of the minimum point of the curve y = x² − 8x + 5. [3 marks]
Click “Check Mark Scheme” to compare your answer, 1) Halve the coefficient of x (−8 ÷ 2 = −4): (x − 4)² − (−4)² + 5 [1 mark]. 2) Simplify: (x − 4)² − 16 + 5 = (x − 4)² − 11 [1 mark]. 3) Minimum point is (4, −11) [1 mark].
Common Exam Questions (3)
What is the general form of a quadratic equation?
The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
When should I use the quadratic formula instead of factorising?
Use the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), when the quadratic expression does not factorise neatly into integer brackets; exam questions typically signal this by asking for solutions rounded to 2 decimal places or 3 significant figures.
What does completing the square tell us about a quadratic graph?
Writing a quadratic in completed-square form, a(x + p)² + q, immediately gives the coordinates of the minimum or maximum turning point (vertex) of the parabola as (-p, q).
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