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Quadratic Formula

HigherHigher tier onlyAQAEdexcelOCR

Practise quadratic formula with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through using the quadratic formula, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. The discriminant b²−4ac tells you how many real solutions there are.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The quadratic formula solves any quadratic equation written in the form \(ax^2 + bx + c = 0\), including those that cannot be factorised.

The formula is \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). You identify \(a\), \(b\) and \(c\) from the equation, substitute them carefully, and evaluate. The \(\pm\) produces the two solutions a quadratic normally has.

Before substituting, the equation must equal zero. If it is written as \(2x^2 + 5x = 3\), you must rearrange to \(2x^2 + 5x - 3 = 0\) first, or every value you substitute will be wrong. Negative coefficients are where most errors creep in, so writing them in brackets is worth the extra second.

Revision notes

Identifying a, b and c

Write the equation in the form \(ax^2 + bx + c = 0\), then read off the three coefficients including their signs.

In \(3x^2 - 5x - 2 = 0\), \(a = 3\), \(b = -5\) and \(c = -2\). Both negatives matter: \(b\) is negative five, not five, and missing this changes the whole answer.

Substituting safely

Put every value in brackets when you substitute, especially negatives. Squaring a negative gives a positive, which is easy to lose.

For \(a = 3\), \(b = -5\), \(c = -2\): \(x = \dfrac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-2)}}{2(3)}\). Notice \(-4ac\) becomes \(+24\) because \(c\) is negative.

The discriminant

The expression under the root, \(b^2 - 4ac\), is called the discriminant and it tells you how many solutions exist.

If it is positive there are two solutions, if zero there is one repeated solution, and if negative there are none at GCSE level. Working it out first is a useful check that you have substituted correctly.

Key points

  • The equation must equal zero before you substitute.
  • \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • Include the sign when reading off \(b\) and \(c\).
  • Use brackets around every substituted value.
  • The discriminant \(b^2 - 4ac\) tells you the number of solutions.
  • The \(\pm\) gives two answers, so give both.

Worked examples

Example 1

Solve \(x^2 + 5x + 3 = 0\), giving answers to 2 decimal places.

Working

\[a = 1,\ b = 5,\ c = 3\]read off the coefficients with their signs
\[b^2 - 4ac = 25 - 12 = 13\]work out the discriminant first
\[x = \frac{-5 \pm \sqrt{13}}{2} = -0.70 \text{ or } -4.30\]substitute and evaluate both roots

Example 2

Solve \(2x^2 - 7x + 3 = 0\).

Working

\[a = 2,\ b = -7,\ c = 3\]note that b is negative
\[b^2 - 4ac = 49 - 24 = 25\]the discriminant is a square number, so the roots are exact
\[x = \frac{7 \pm 5}{4} = 3 \text{ or } \tfrac{1}{2}\]evaluate both roots

Example 3

Solve \(3x^2 + 2x = 8\).

Working

\[3x^2 + 2x - 8 = 0\]rearrange so the equation equals zero before substituting
\[b^2 - 4ac = 4 + 96 = 100\]with a = 3, b = 2, c = −8
\[x = \frac{-2 \pm 10}{6} = \tfrac{4}{3} \text{ or } -2\]evaluate both roots

Common mistakes

  • Substituting before rearranging to equal zero.

    In 3x² + 2x = 8 the value of c is −8, not 8. The equation must be set to zero first.

  • Losing the sign of b.

    For 2x² − 7x + 3, b is −7, so −b is +7. Writing −7 in the numerator gives the wrong roots.

  • Making \(b^2\) negative.

    Squaring −5 gives +25. Brackets around the substituted value prevent this.

  • Dividing only part of the numerator by 2a.

    The whole of −b ± √(b² − 4ac) is divided by 2a. The fraction line acts as a bracket.

Exam tips

  • Work out the discriminant on its own line before substituting into the full formula.
  • Use the formula when factorising is not obvious, rather than spending time hunting for factors.
  • Give both roots unless the context rules one out, such as a negative length.
  • Round only at the end, and to the accuracy the question asks for.

Key terms

Quadratic formula
The formula that solves any quadratic equation in the form \(ax^2 + bx + c = 0\).
Discriminant
The expression \(b^2 - 4ac\), which determines the number of solutions.
Root
A solution of an equation, where the curve crosses the x-axis.
Coefficient
The number multiplying a term, such as the 3 in \(3x^2\).

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.