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Algebra

Algebra

Equations that lead to quadratics

Quadratic Equation Solving Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Get it to = 0
  4. Quadratics in disguise
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.5: equations that lead to quadratics
  • 2.5Turn equations with brackets, squares or fractions into \(ax^{2}+bx+c=0\), then solve.
  • 2.5Solve fractional equations such as \(\dfrac2{x+1}+\dfrac3{x+3}=1\).
  • 2.5Pick factorising or the formula, and give answers to the accuracy asked.

Before you start: Solving by factorising, Algebraic fractions and the quadratic formula.

Formula bank

Quadratic formulaOn the formula sheet\(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)for \(ax^{2}+bx+c=0\).Use it when it won't factorise, or 'give answers to 2 d.p.'.
Clearing two fractionsLearn it\(\dfrac A{P}+\dfrac B{Q}=C\ \Rightarrow\ AQ+BP=CPQ\)multiply EVERY term by \(PQ\).Use it when x in the denominators.

Get it to = 0

Every equation on this page becomes 'quadratic = 0'. The work is getting it there without dropping a term or a sign.

General methodHow to solve an equation that leads to a quadratic

  1. Clear any fractions: multiply EVERY term by the product of the denominators (keep brackets).
  2. Expand every bracket.
  3. Move everything to one side so it reads \(ax^{2}+bx+c=0\) (make the \(x^{2}\) term positive).
  4. Solve: factorise if you can; if not, or if they ask for decimals, use the formula.
  5. Check no answer makes a denominator zero.

Example Solve \((x+2)^{2}=3x+10\).

Expand the square, move everything to the left, factorise.

  1. \(x^{2}+4x+4=3x+10\)
  2. \(x^{2}+x-6=0\)
  3. \((x+3)(x-2)=0\)

Answer \(x=-3\) or \(x=2\)

Example 2 Solve \(\dfrac2{x+1}+\dfrac3{x+3}=1\).

Multiply every term by \((x+1)(x+3)\).

  1. \(2(x+3)+3(x+1)=(x+1)(x+3)\)
  2. \(5x+9=x^{2}+4x+3\)
  3. \(x^{2}-x-6=0\)
  4. \((x-3)(x+2)=0\)

Answer \(x=3\) or \(x=-2\)

Try it yourself

1
Solve \(x(x-3)=28\).
Show answer

Answer 7, −4

2
Solve \((x-1)^{2}=16\).
Hint
\(x-1=\pm4\).
Show answer

Answer 5, −3

3
Solve \(\dfrac{12}x=x+1\).
Hint
× x: \(12=x^{2}+x\).
Show answer

Answer 3, −4

Quadratics in disguise

\(x^{4}-13x^{2}+36=0\) is a quadratic in \(x^{2}\): call \(x^{2}=u\), solve \(u^{2}-13u+36=0\), then square-root each answer (±).

Example Solve \(x^{4}-5x^{2}+4=0\).

Factorise as a quadratic in \(x^{2}\).

  1. \((x^{2}-1)(x^{2}-4)=0\)
  2. \(x^{2}=1\) or \(x^{2}=4\)

Answer \(x=\pm1\) or \(\pm2\)

Ways they test it

Type 1 Brackets and squares

"Solve \((x+2)^{2}=3x+10\)." (3 marks)

Spot it a squared bracket or x times a bracket.

  1. Expand.
  2. Make it = 0.
  3. Factorise.

Type 2 Fractions, answers exact

"Solve \(\dfrac2{x+1}+\dfrac3{x+3}=1\)." (5 marks)

Spot it x in two denominators, no decimals asked.

  1. × by both denominators.
  2. Expand and simplify to = 0.
  3. Factorise.

Type 3 Fractions, answers to 2 d.p.

"Solve \(\dfrac3{x-2}-\dfrac2{x+1}=1\). Give your answers to 2 decimal places." (6 marks)

Spot it '2 decimal places' = the formula.

  1. Clear the fractions (careful with the minus).
  2. Quadratic = 0.
  3. Formula; round at the end.

For example \(3(x+1)-2(x-2)=(x-2)(x+1)\)

  1. \(x+7=x^{2}-x-2\)
  2. \(x^{2}-2x-9=0\)
  3. \(x=\dfrac{2\pm\sqrt{40}}2\)

Answer 4.16 or −2.16

Type 4 Hidden quadratic

"Solve \(x^{4}-13x^{2}+36=0\)."

Spot it powers \(x^{4}\) and \(x^{2}\) only.

  1. Treat \(x^{2}\) as one letter.
  2. Square-root each answer (±).

For example \((x^{2}-4)(x^{2}-9)=0\)

    Answer \(\pm2,\ \pm3\)

    Mistakes to avoid

    • ✗ \((x+2)^{2}=x^{2}+4\)✓ \(x^{2}+4x+4\)There is a middle term.
    • ✗ \(-2(x-2)=-2x-4\)✓ \(-2x+4\)− × − = +.
    • ✗ Multiplying only the fractions by the denominators✓ multiply every term, including the right-hand sideKeep the balance.
    • ✗ \(x^{2}=4\Rightarrow x=2\)✓ \(x=\pm2\)Two square roots.

    Summary

    • Clear fractions, expand, make = 0, solve.
    • Factorise if you can; otherwise the formula (round at the end).
    • Quartic with only \(x^{4}\), \(x^{2}\): a quadratic in \(x^{2}\).
    • Check no answer makes a denominator zero.

    If you see …

    • x in a denominatormultiply every term by that denominator.
    • "give your answers to 2 decimal places"use the quadratic formula.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Solve \(x^{2}=6x\).
    Hint
    \(x^{2}-6x=0\); don't divide by x.
    Show answer

    Answer 0, 6

    2
    Solve \(\dfrac6x=x-1\).
    Hint
    × x: \(6=x^{2}-x\).
    Show answer

    Answer 3, −2

    3
    Solve \((x+2)^{2}=3x+10\).
    Show answer

    Answer −3, 2

    Medium

    1
    Solve \(\dfrac2{x+1}+\dfrac3{x+3}=1\).
    Guide me

    What is it asking? x

    What do we know? × \((x+1)(x+3)\)

    Plan \(x^{2}-x-6=0\)

    Show working
    1. \(2(x+3)+3(x+1)=(x+1)(x+3)\)
    2. \(x^{2}-x-6=0\)
    3. \((x-3)(x+2)=0\)

    Answer 3, −2

    2
    Solve \(\dfrac4{x+2}+\dfrac1{x-1}=1\).
    Guide me

    What is it asking? x

    What do we know? × \((x+2)(x-1)\)

    Plan \(x^{2}-4x=0\)

    Show working
    1. \(4(x-1)+(x+2)=(x+2)(x-1)\)
    2. \(5x-2=x^{2}+x-2\)
    3. \(x^{2}-4x=0\)

    Answer 0, 4

    3
    Solve \(x^{4}-13x^{2}+36=0\).
    Guide me

    What is it asking? four values

    What do we know? quadratic in \(x^{2}\)

    Plan \((x^{2}-4)(x^{2}-9)=0\)

    Show working
    1. \(x^{2}=4\) or \(9\)
    2. \(x=\pm2\) or \(\pm3\)

    Answer \(\pm2,\ \pm3\)

    Hard

    1
    Solve \(\dfrac2{x-1}+\dfrac1{x+2}=1\), giving your answers to 2 decimal places.
    Guide me

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

    What do we know? × \((x-1)(x+2)\)

    Plan \(x^{2}-2x-5=0\), then the formula

    Show working
    1. \(2(x+2)+(x-1)=(x-1)(x+2)\)
    2. \(3x+3=x^{2}+x-2\)
    3. \(x^{2}-2x-5=0\)
    4. \(x=\dfrac{2\pm\sqrt{24}}2\)

    Answer 3.45, −1.45

    PartAnswer / workingMarksGuidance
    \(2(x+2)+(x-1)=(x-1)(x+2)\)M2
    \(x^{2}-2x-5=0\)A1
    \(\dfrac{2\pm\sqrt{24}}2\)M1
    3.45, −1.45A2
    Total: 6 marks
    2
    Solve \(\dfrac3{x-2}-\dfrac2{x+1}=1\), giving your answers to 2 decimal places.
    Guide me

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

    What do we know? × \((x-2)(x+1)\); watch \(-2(x-2)\)

    Plan \(x^{2}-2x-9=0\)

    Show working
    1. \(3(x+1)-2(x-2)=(x-2)(x+1)\)
    2. \(x+7=x^{2}-x-2\)
    3. \(x^{2}-2x-9=0\)
    4. \(x=\dfrac{2\pm\sqrt{40}}2\)

    Answer 4.16, −2.16

    PartAnswer / workingMarksGuidance
    \(3(x+1)-2(x-2)=(x-2)(x+1)\)M2
    \(x^{2}-2x-9=0\)A1
    \(\dfrac{2\pm\sqrt{40}}2\)M1
    4.16, −2.16A2
    Total: 6 marks
    3
    Solve \(\dfrac5{x+2}+\dfrac2x=3\).
    Guide me

    What is it asking? x

    What do we know? × \(x(x+2)\)

    Plan \(3x^{2}-x-4=0\)

    Show working
    1. \(5x+2(x+2)=3x(x+2)\)
    2. \(7x+4=3x^{2}+6x\)
    3. \(3x^{2}-x-4=0\)
    4. \((3x-4)(x+1)=0\)

    Answer \(\tfrac43,\ -1\)

    PartAnswer / workingMarksGuidance
    \(5x+2(x+2)=3x(x+2)\)M1
    \(3x^{2}-x-4=0\)A1
    \((3x-4)(x+1)=0\)M1
    \(\tfrac43,\ -1\)A1
    Total: 4 marks

    Can I do this?

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

    Quiz

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