At A-Level, solving quadratics goes beyond factorising. You need exact roots from the quadratic formula, and the ability to spot a quadratic hidden inside another equation: a quartic in x², an equation in √x, or an exponential equation in 2ˣ or 3ˣ. Substitute, solve the quadratic, then convert back and reject any value that is impossible.
Solving quadratics
- Factorise: (px + q)(rx + s) = 0 gives x = −q/p or x = −s/r
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
- Quartic: let u = x²
- Exponential: let y = bˣ, so b²ˣ = y²
Always rearrange to = 0 before factorising.
Factorising and solving
Rearrange to ax² + bx + c = 0, factorise, and set each bracket to zero. When a ≠ 1, look for two numbers that multiply to ac and add to b, and use them to split the middle term. Everything must be on one side first: a quadratic must equal zero before you can use the brackets.
The quadratic formula and exact roots
When a quadratic does not factorise, use the formula and leave the answer exact. Simplify the surd fully, for example √40 = 2√10, and cancel any common factor, so that answers tidy to a form like (2 ± √10)/2.
Hidden quadratics
Many equations turn into quadratics after a substitution. x⁴ − 5x² + 4 = 0 is a quadratic in x²; an equation containing x and √x is a quadratic in √x; and an equation with fractions often becomes a quadratic once you multiply through. After solving, convert back and check each value: √x can never be negative, so a negative root for √x must be rejected.
Exponential equations that are quadratics
An equation such as 9ˣ − 4 × 3ˣ + 3 = 0 is a hidden quadratic. Let y = 3ˣ; then 9ˣ = (3²)ˣ = (3ˣ)² = y². Solve the quadratic in y, then solve 3ˣ = y for each root. Since bˣ is always positive, any negative value of y gives no solution.
Where a line meets a curve
To find where a line meets a curve, substitute the line into the curve's equation to get a single quadratic, solve it for x, then substitute each x back into the line to find y. The discriminant of that quadratic tells you how many intersection points there are, and a discriminant of zero means the line is a tangent.
Worked examples
Example 1. Solve x² − 7x + 12 = 0.
- Two numbers that multiply to 12 and add to −7 are −3 and −4.
- (x − 3)(x − 4) = 0.
Answer: x = 3 or x = 4.
Marks: M1 for factorising, A1 for both roots.
Example 2. Solve 2x² − 4x − 3 = 0, giving exact answers.
- a = 2, b = −4, c = −3: x = (4 ± √(16 + 24))/4 = (4 ± √40)/4.
- √40 = 2√10, so x = (4 ± 2√10)/4.
- Cancel the common factor of 2.
Answer: x = (2 ± √10)/2.
Marks: M1 for the formula, M1 for simplifying √40, A1 for the answer.
Example 3. Solve x⁴ − 5x² + 4 = 0.
- Let u = x²: u² − 5u + 4 = 0.
- (u − 1)(u − 4) = 0, so u = 1 or u = 4.
- x² = 1 gives x = ±1; x² = 4 gives x = ±2.
Answer: x = ±1, ±2.
Marks: M1 for the quadratic in u, M1 for u = 1, 4, A1 for all four roots.
Example 4. Solve 9ˣ − 4 × 3ˣ + 3 = 0.
- Let y = 3ˣ, so 9ˣ = y²: y² − 4y + 3 = 0.
- (y − 1)(y − 3) = 0, so y = 1 or y = 3.
- 3ˣ = 1 gives x = 0; 3ˣ = 3 gives x = 1.
Answer: x = 0 or x = 1.
Marks: M1 for the quadratic in y, M1 for y = 1, 3, A1 for both values of x.
Example 5. Find where y = x + 3 meets y = x² + 1.
- Set equal: x² + 1 = x + 3, so x² − x − 2 = 0.
- (x − 2)(x + 1) = 0, so x = 2 or x = −1.
- From the line: x = 2 gives y = 5; x = −1 gives y = 2.
Answer: (2, 5) and (−1, 2).
Marks: M1 for the quadratic, M1 for both x-values, A1 for both points.
Common mistakes
- Factorising before rearranging. Move every term to one side so the quadratic equals zero.
- Not simplifying the surd. √40 should be written as 2√10, then any common factor cancelled.
- Stopping at u. After substituting u = x², convert back: u = 4 gives x = ±2.
- Keeping an impossible value. 3ˣ and √x are never negative; reject any negative solution for them.
- Forgetting the y-coordinates. Intersection points need both coordinates, found from the line.
How solving quadratics is examined at A-Level
Solving quadratics turns up across the WMA11 paper (Pure 1, Edexcel International A-Level) and throughout Year 1 of Edexcel A-Level (9MA0). Expect questions that:
- ask for exact roots, simplified fully
- disguise a quadratic as a quartic, a surd equation or an exponential equation
- ask for the points where a line meets a curve
- use “show that … can be written as” before “hence solve”
Work through the Pure 1 quadratics exam questions and check each step against the mark scheme, then try the Pure 1 quadratics worksheet. The algebra revision engine generates unlimited extra equations.
Next, see completing the square and the discriminant. For the GCSE introduction, see quadratic equations.
Frequently asked questions
How do you know if a quadratic will factorise?
If b² − 4ac is a perfect square, the quadratic factorises with whole numbers or fractions. Otherwise, use the formula or complete the square.
What is a hidden quadratic?
An equation that becomes a quadratic after a substitution, such as x⁴ − 5x² + 4 = 0 with u = x², or 9ˣ − 4 × 3ˣ + 3 = 0 with y = 3ˣ.
Why do some solutions get rejected?
Because the original variable cannot take certain values. For example, 3ˣ is always positive, so 3ˣ = −2 has no solution.