Practice Maths

The Quadratic Formula

Key Ideas

Key Terms

quadratic formula
Solves any equation of the form ax² + bx + c = 0 (where a ≠ 0).
discriminant
Δ = b² − 4ac tells you how many solutions exist before you do the full calculation.
surd
An irrational square root left in exact form (e.g. √7, 2√3). It is more precise than a decimal approximation.
x  =  −b  ±  √(b² − 4ac) 2a
Discriminant Δ = b² − 4ac Number of solutions Type
Δ > 0Two distinct real solutionsRational if Δ is a perfect square; surd otherwise
Δ = 0One solution (repeated root)Always rational: x = −b ÷ (2a)
Δ < 0No real solutionsParabola does not cross the x-axis
Step Action Example: x² − 5x + 2 = 0
1Standard form (= 0)Already done: x² − 5x + 2 = 0
2Identify a, b, ca = 1, b = −5, c = 2
3Calculate Δ = b² − 4acΔ = 25 − 8 = 17  (> 0, so two solutions)
4Apply formula & simplifyx = (5 ± √17) / 2   (cannot simplify further)
Hot Tip — Sign Errors on b! When b is negative (e.g. b = −5), the formula gives −b = −(−5) = +5 in the numerator. This is the most common error. Always write −b explicitly before substituting numbers.

Worked Example 1 — Two surd solutions

Solve x² − 6x + 7 = 0.

Identify: a = 1, b = −6, c = 7.

Δ = (−6)² − 4(1)(7) = 36 − 28 = 8  (Δ > 0 ⇒ two solutions; 8 is not a perfect square ⇒ surd answers)

x = (6 ± √8) / 2 = (6 ± 2√2) / 2 = 3 ± √2

Simplification: √8 = √(4 × 2) = 2√2, then divide all of (6 ± 2√2) by 2.

Worked Example 2 — One repeated solution

Solve 4x² − 12x + 9 = 0.

Identify: a = 4, b = −12, c = 9.

Δ = 144 − 144 = 0  (Δ = 0 ⇒ exactly one solution)

x = 12 / (2 × 4) = 12 / 8 = 3/2

Why We Need the Quadratic Formula

Factorising is fast and elegant — but it only works neatly when the roots happen to be integers or simple fractions. The equation x² − 6x + 7 = 0 has no integer factors, yet it has two perfectly valid real solutions (3 + √2 and 3 − √2). The quadratic formula always works, for any quadratic, regardless of how messy the solutions are.

In fact, factorising is just a special case: when Δ is a perfect square, the formula gives rational answers that could also have been found by factorising. The formula subsumes factorising — it is the more powerful tool.

Choosing the Right Method

Use factorising when the equation looks like it has nice integer roots — try it for 30 seconds. If it doesn’t come together quickly, switch to the formula. A good rule of thumb:

  • Calculate Δ first. If Δ is a small perfect square (1, 4, 9, 16, 25, …), factorising will be quick.
  • If Δ is not a perfect square, the roots are irrational — you must use the formula (or complete the square).
  • If the question specifically asks for an exact answer or a decimal approximation to a given number of places, use the formula.

Identifying a, b, c Carefully

The formula requires the equation in standard form: ax² + bx + c = 0. Before reading off a, b, and c:

  • Rearrange first. 2x² = 3x − 1 becomes 2x² − 3x + 1 = 0, giving a = 2, b = −3, c = 1.
  • Watch the sign of b. In 5x² − 7x + 2 = 0, b = −7 (negative). The formula uses −b = +7.
  • Missing terms. In x² − 9 = 0, there is no x term, so b = 0. In 3x² + 4x = 0, there is no constant, so c = 0.
  • Leading negative. If the x² coefficient is negative (e.g. −x² + 3x + 10 = 0), multiply through by −1 to get x² − 3x − 10 = 0 before proceeding.

Simplifying Surd Answers

After applying the formula you often have a surd in the numerator. Simplify √Δ as far as possible before dividing by 2a — it often allows further cancellation.

Example: Solve x² − 4x − 1 = 0. Δ = 16 + 4 = 20. √20 = √(4 × 5) = 2√5. So x = (4 ± 2√5) / 2 = 2 ± √5. If you don’t simplify first, you get (4 ± √20)/2 — technically correct but not fully simplified.

Critical rule: you can only cancel a factor from the numerator if it divides every term — both the −b part and the ±√Δ part. In (6 ± 2√3)/2, the factor 2 divides 6 and 2√3, so the result is 3 ± √3. In (5 ± √3)/2, there is no common factor, so this is already fully simplified.

Exact Form vs Decimal Approximation

Exact form (leaving surds unsimplified as decimals) is mathematically precise and preferred in algebraic questions, geometry proofs, and any time the question asks to “give an exact answer”.

Decimal approximation is appropriate for real-world contexts (finding a length, a time, a cost) or when the question specifies rounding. Use your calculator for √Δ and round only at the final step.

Common Mistakes Checklist:
  1. Forgetting the ± and writing only one solution.
  2. Computing −b incorrectly when b is negative (the double negative).
  3. Dividing only part of the numerator: (6 ± √8) / 2 ≠ 3 ± √8 — you must also divide √8 by 2.
  4. Failing to simplify the surd: √8 = 2√2, so (6 ± √8)/2 = 3 ± √2, not 3 ± √8/2.
  5. Not rearranging to standard form first.

Mastery Practice

  1. For each quadratic equation, identify a, b, and c, then calculate the discriminant Δ = b² − 4ac. Do not solve yet. Fluency

      Equation (in standard form) a b c Δ = b² − 4ac
    (a)x² + 5x + 6 = 0    
    (b)2x² − 3x − 5 = 0    
    (c)x² − 4x + 4 = 0    
    (d)3x² + 2x + 1 = 0    

  2. Apply the quadratic formula to solve each equation. All answers are rational. Fluency

     EquationSolutions
    (a)x² − 5x + 6 = 0 
    (b)x² + x − 12 = 0 
    (c)2x² − 7x + 3 = 0 
    (d)3x² + 5x − 2 = 0 

  3. Apply the formula and give each answer in exact simplified surd form. Fluency

    1. x² − 4x + 1 = 0
    2. x² + 6x + 3 = 0
    3. 2x² − 2x − 1 = 0
    4. x² − 8x + 3 = 0

  4. Without solving, use the discriminant to state (i) the number of real solutions and (ii) whether they are rational or irrational. Fluency

    1. x² − 6x + 9 = 0
    2. x² − 4x + 5 = 0
    3. x² − 6x + 7 = 0
    4. 2x² − 7x + 3 = 0

  5. For each equation, calculate the discriminant, state whether you would use factorising or the quadratic formula, then solve. Understanding

    1. x² − 9x + 20 = 0
    2. x² − 3x − 1 = 0
    3. 2x² + 5x − 12 = 0
    4. x² − 2x − 7 = 0

  6. Projectile motion. Understanding

    Physics. A ball is thrown upward from a platform. Its height above the ground (in metres) after t seconds is given by h = −5t² + 20t + 25.
    1. What is the initial height of the ball (at t = 0)?
    2. Find the time(s) when the ball is at a height of 40 m.
    3. Find when the ball hits the ground (h = 0). Which solution must be rejected and why?

  7. Diagonal of a rectangle. Understanding

    Geometry. A rectangle has a width of x cm and a length of (x + 3) cm. Its diagonal measures 9 cm.
    1. Use Pythagoras’ theorem to write a quadratic equation in standard form.
    2. Solve using the quadratic formula, giving the answer in exact surd form. (Discard any solution that is not physically valid.)
    3. State the width and length of the rectangle, rounded to 2 decimal places.

  8. Discriminant with a parameter. Understanding

    Algebraic analysis. Consider the quadratic equation x² − kx + (k + 3) = 0, where k is a real constant.
    1. Write an expression for the discriminant Δ in terms of k, then fully factorise it.
    2. Find the values of k for which the equation has exactly one solution.
    3. Find the values of k for which the equation has two distinct real solutions.
    4. When k = 8, solve the equation, giving your answer in exact surd form.

  9. Rectangular fencing. Problem Solving

    Agriculture. A farmer has 80 m of fencing to enclose a rectangular paddock against a straight barn wall. The wall forms one complete side, so no fencing is needed there. The enclosed area must be exactly 750 m².
    1. Let the width of the paddock (perpendicular to the wall) be w metres. Write an expression for the length in terms of w, then form a quadratic equation for the area.
    2. Solve the equation using the quadratic formula. Give exact answers first, then round to 2 decimal places.
    3. State both possible sets of dimensions (width × length) and verify that each has the correct area.

  10. A number and its reciprocal. Problem Solving

    Number theory. The sum of a positive number and its reciprocal is 29/10.
    1. Let the number be x. Write the equation x + 1/x = 29/10 in standard quadratic form (multiply through to clear fractions).
    2. Apply the quadratic formula to find both values of x. Are the solutions rational or irrational?
    3. Verify both solutions satisfy the original condition x + 1/x = 29/10.
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