PrepWithTee owl logo PrepWithTee
Algebra

Algebra

The quadratic formula

Quadratic Formula Extended 5 min read

On this page
  1. In this note
  2. Formula bank
  3. Using the formula
  4. Ways they test it
  5. Mistakes to avoid
  6. Summary
  7. Practice
  8. Can I do this?
  9. Quiz

In this note

Syllabus 2.5: the quadratic formula
  • 2.5Solve \(ax^{2}+bx+c=0\) with the quadratic formula, giving answers to 2 d.p. or 3 s.f.
  • 2.5Give exact answers like \(3\pm\sqrt5\).
  • 2.5Rearrange first and use the signs of a, b and c correctly.

Before you start: Completing the square (the formula comes from it). When it factorises, factorising is quicker: Solving by factorising.

Formula bank

Quadratic formulaOn the formula sheet\(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)for \(ax^{2}+bx+c=0\); the WHOLE top is divided by \(2a\).Use it when it won't factorise, or decimals are asked for.

Using the formula

The formula solves ANY quadratic. It is completing the square done once with letters. If the number under the square root, \(b^{2}-4ac\), is negative, the equation has no solutions.

General methodHow to use the quadratic formula

  1. Rearrange to \(ax^{2}+bx+c=0\).
  2. Write down a, b and c WITH their signs.
  3. Work out \(b^{2}-4ac\) first (use brackets: \((-3)^{2}=9\)).
  4. Put it into \(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) and work out the + answer and the − answer.
  5. Round only at the end, to the accuracy asked.

Example Solve \(2x^{2}+5x-4=0\), giving your answers to 2 decimal places.

\(a=2,\ b=5,\ c=-4\).

  1. \(b^{2}-4ac=25+32=57\)
  2. \(x=\dfrac{-5\pm\sqrt{57}}4\)
  3. \(x=\dfrac{-5+7.550}4=0.638\) or \(x=\dfrac{-5-7.550}4=-3.138\)

Answer 0.64 or −3.14

Example 2 Solve \(x^{2}-6x+4=0\), giving exact answers.

\(a=1,\ b=-6,\ c=4\), so \(-b=6\).

  1. \(b^{2}-4ac=36-16=20\)
  2. \(x=\dfrac{6\pm\sqrt{20}}2=\dfrac{6\pm2\sqrt5}2\)

Answer \(3\pm\sqrt5\)

Try it yourself

1
\(a=2,\ b=-3,\ c=-6\). Work out \(b^{2}-4ac\).
Show answer

Answer 57

2
Solve \(x^{2}+3x-5=0\) to 2 d.p.
Show answer

Answer 1.19, −4.19

3
Solve \(4x^{2}-2x-1=0\) to 2 d.p.
Show answer

Answer 0.81, −0.31

Ways they test it

Type 1 Answers to 2 d.p.

"Solve \(2x^{2}-3x-6=0\). Show all your working and give your answers correct to 2 decimal places." (4 marks)

Spot it '2 decimal places' or '3 significant figures'.

  1. a, b, c with signs.
  2. Formula with brackets.
  3. Both answers, rounded at the end.

For example \(x=\dfrac{3\pm\sqrt{57}}4\)

    Answer 2.64, −1.14

    Type 2 Rearrange first

    "Solve \(5x^{2}=3x+1\)."

    Spot it not '= 0' yet.

    1. Move everything left: \(5x^{2}-3x-1=0\).
    2. Then the formula.

    For example \(x=\dfrac{3\pm\sqrt{29}}{10}\)

      Answer 0.84, −0.24

      Type 3 Exact answers

      "Solve \(x^{2}-4x-1=0\), giving your answers in the form \(p\pm\sqrt q\)." (non-calculator)

      Spot it 'exact' or 'surd form'.

      1. Formula.
      2. Simplify the surd, divide every term by 2a.

      For example \(x=\dfrac{4\pm\sqrt{20}}2\)

      1. \(=\dfrac{4\pm2\sqrt5}2\)

      Answer \(2\pm\sqrt5\)

      Type 4 The last step of a bigger question

      "Show that ... = 0. Solve ... giving answers to 2 d.p." (a problem or a fraction equation)

      Spot it after forming an equation.

      1. Use the given quadratic even if you couldn't 'show' it.

      Mistakes to avoid

      • ✗ \(b=-3\): \(b^{2}=-9\)✓ \((-3)^{2}=9\)Square with brackets.
      • ✗ \(b=-6\): \(-b=-6\)✓ \(-b=+6\)Minus a negative is positive.
      • ✗ \(-b\pm\dfrac{\sqrt{b^{2}-4ac}}{2a}\)✓ \(\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)Divide the WHOLE top by 2a.
      • ✗ Rounding before the end✓ keep full calculator valuesEarly rounding changes the 2nd decimal place.

      Summary

      • \(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(ax^{2}+bx+c=0\).
      • a, b, c with their signs; brackets when substituting.
      • Two answers: one with +, one with −.
      • \(b^{2}-4ac<0\): no solutions.

      If you see …

      • "give your answers to 2 decimal places"the quadratic formula.
      • "exact" / "surd form"formula, then simplify the surd.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      Solve \(x^{2}+7x+3=0\) to 2 d.p.
      Show answer

      Answer −0.46, −6.54

      2
      Solve \(3x^{2}+x-1=0\) to 2 d.p.
      Show answer

      Answer 0.43, −0.77

      3
      How many solutions does \(x^{2}+x+3=0\) have?
      Hint
      \(b^{2}-4ac=1-12\).
      Show answer

      Answer none

      Medium

      1
      Solve \(2x^{2}-3x-6=0\) to 2 d.p.
      Guide me

      What is it asking? x to 2 d.p.

      What do we know? \(a=2,\ b=-3,\ c=-6\)

      Plan \(b^{2}-4ac=57\)

      Show working
      1. \(x=\dfrac{3\pm\sqrt{57}}4\)
      2. 2.637…, −1.137…

      Answer 2.64, −1.14

      2
      Solve \(5x^{2}=3x+1\) to 2 d.p.
      Guide me

      What is it asking? x to 2 d.p.

      What do we know? rearrange first

      Plan \(5x^{2}-3x-1=0\)

      Show working
      1. \(x=\dfrac{3\pm\sqrt{29}}{10}\)

      Answer 0.84, −0.24

      3
      Solve \(3x^{2}-2x-7=0\) to 2 d.p.
      Guide me

      What is it asking? x to 2 d.p.

      What do we know? \(a=3,\ b=-2,\ c=-7\)

      Plan \(4+84=88\)

      Show working
      1. \(x=\dfrac{2\pm\sqrt{88}}6\)

      Answer 1.90, −1.23

      Hard

      1
      Solve \(x^{2}-4x-1=0\). Give your answers in the form \(p\pm\sqrt q\).
      Guide me

      What is it asking? exact x

      What do we know? \(b^{2}-4ac=20\)

      Plan \(\sqrt{20}=2\sqrt5\), divide by 2

      Show working
      1. \(x=\dfrac{4\pm\sqrt{20}}2\)
      2. \(=2\pm\sqrt5\)

      Answer \(2\pm\sqrt5\)

      PartAnswer / workingMarksGuidance
      \(\dfrac{4\pm\sqrt{20}}2\)M1
      \(2\pm\sqrt5\)A1
      Total: 2 marks
      2
      Solve \(2x^{2}+x-5=0\) to 2 d.p.
      Guide me

      What is it asking? x to 2 d.p.

      What do we know? \(b^{2}-4ac=1+40\)

      Plan \(\sqrt{41}\)

      Show working
      1. \(x=\dfrac{-1\pm\sqrt{41}}4\)

      Answer 1.35, −1.85

      PartAnswer / workingMarksGuidance
      \(\sqrt{41}\)B1
      \(\dfrac{-1\pm\sqrt{41}}4\)M1
      1.35, −1.85A1A1
      Total: 4 marks
      3
      Solve \(x(x+5)=1-2x\) to 2 d.p.
      Guide me

      What is it asking? x to 2 d.p.

      What do we know? expand and rearrange

      Plan \(x^{2}+7x-1=0\)

      Show working
      1. \(x^{2}+5x=1-2x\)
      2. \(x^{2}+7x-1=0\)
      3. \(x=\dfrac{-7\pm\sqrt{53}}2\)

      Answer −0.14, −7.14

      PartAnswer / workingMarksGuidance
      \(x^{2}+7x-1=0\)M1
      \(\dfrac{-7\pm\sqrt{53}}2\)M1
      −0.14, −7.14A1A1
      Total: 4 marks

      Can I do this?

      Anything unticked: revise that section first. All ticked: take the quiz.

      Quiz

      The multiple-choice and guided questions need JavaScript. The concept checks and exam practice below work without it.

      Your progress on this note is saved when JavaScript is on.