Solving Quadratic Equations by Factors

Solving by Factorisation

Solving by Factorisation

Consider the following steps to solve quadratic equations by factorisation.

Step $0$: Read the question.
  • $x^2=3x-2$
  • Step $1$: Rearrange the equation so the right-hand side is zero, if necessary.
  • $x^2-3x+2=0$
  • Step $2$: Fully factorise the other side.
  • $(x-1)(x-2)=0$
  • Step $3$: Use the Null Factor law; $a\times b=0$ means $a=0$ or $b=0$.
  • $x-1=0$ or $x-2=0$
  • Step $4$: Solve the resulting linear equations.
  • $x=1$ or $x=2$

  • Watch out the following case carefully!
    Solve $x^2 = 3x$.
    \( \begin{align} \displaystyle x^2 &= 3x \\ \dfrac{x^2}{x} &= \dfrac{3x}{x} \\ \therefore x &= 3 \\ \end{align} \)
    The solution above is not correct, as we lose the solution $x=0$ by dividing both sides by $x$.

    Correct solution should be;
    \( \begin{align} \displaystyle x^2 &= 3x \\ x^2 -3x &= 0 \\ x(x-3) &= 0 \\ x &= 0 \text{ or } x-3 = 0 \\ \therefore x &= 0 \text{ or } 3 \\ \end{align} \)

    Example 1

    Solve $x^2 - 4x =0$ for $x$.

    Example 2

    Solve $x^2 = 3x+4$ for $x$.

    Example 3

    Solve $x^2 = 4x-4$ for $x$.

    Example 4

    Solve $2x^2 = -x + 3$ for $x$.

    Example 5

    Solve $ \displaystyle 3x + \dfrac{2}{x} = -7$ for $x$.

    Related Video Lessons:
    Solving Quadratics by Factors
    Quadratic Equations in Fractions