# Solving Quadratic Equations by Factors

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 $x$.

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