MathematicsExam code: 0580

Quadratic Equations

Algebra and Sequences

The formula is printed in the List of formulas at the front of the paper, so it does not have to be memorised
For ax² + bx + c = 0 with a not zero, x = (−b ± √(b² − 4ac)) ÷ 2a
Read a, b and c off the rearranged equation, taking the sign in front of each with it
Write the substituted line out in full before evaluating anything, because that line is where the method marks sit
Put brackets around every negative value as you substitute, since −b with b negative is a positive number
Work the + and the − versions separately to get the two solutions

Worked example

Answers to two decimal places

Solve 3x² − 7x − 5 = 0. Give each answer correct to 2 decimal places.

Solution:

The expression does not factorise, and the answers are wanted as decimals, so use the formula printed on page 2 of the paper
Read the three coefficients off the equation with their signs: a = 3, b = −7, c = −5
Substitute, putting brackets round every negative so no sign is lost
x = (−(−7) ± √((−7)² − 4 × 3 × (−5))) ÷ (2 × 3)
Write that whole line down before touching the calculator: a mark is given for the value under the root and another for the full substitution
Under the root: (−7)² = 49, and −4 × 3 × (−5) = +60, so the value is 49 + 60 = 109
x = (7 ± √109) ÷ 6, and √109 = 10.4403…
The ± gives two answers, and both are wanted
Plus: x = (7 + 10.4403…) ÷ 6 = 2.9067…
Minus: x = (7 − 10.4403…) ÷ 6 = −0.5734…
Rounded as asked: x = 2.91 or x = −0.57
Keep the full accuracy until this last line: rounding √109 to 10.4 first gives (7 + 10.4) ÷ 6 = 2.90, which is wrong

When the question asks for an exact answer, stop before rounding and simplify the surd instead
Take out the largest square factor of the number under the root, then cancel any factor shared by every part of the fraction
A number under the root that is a perfect square means the quadratic would have factorised, which is a useful check
Solutions such as (7 ± √109) ÷ 6 are already exact and need no decimal at all

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