Solving Quadratic Equations by Factorising: Quick Revision

A brief note on Solving Quadratic Equations by Factorising. Read it before a test. Try each question at the end before you open its answer.

Quadratic equations

• Standard form: $ax^2 + bx + c = 0$, with $a \neq 0$. The numbers $b$ and $c$ may be $0$.
• Expand and collect every term on one side before you decide. In $(x + 4)^2 = x^2 + 2x$ the $x^2$ terms cancel, so it is linear.
• A root makes the left side $0$. Test it by putting it in.
• A quadratic equation has at most two roots, where its curve meets the $x$ axis.

Solving by factorisation

• If a product is $0$, one of its factors is $0$.
• Split the middle term: find $p$ and $q$ with $p + q = b$ and $p \times q = a \times c$, write $bx = px + qx$, and group.
• Each factor gives a root: $(2x - 3) = 0$ gives $x = \dfrac{3}{2}$.
• Special cases: no constant term, take out $x$; a difference of squares; a perfect square gives a repeated root.

Word problems

• Name the unknown with its unit, write the equation, solve, and check the meaning.
• Keep only the roots that fit: lengths, speeds, ages and counts are not negative.
• Time $=$ distance $\div$ speed. Clear fractions by multiplying by every denominator.

A worked example

Solve $3x^2 - 5x - 2 = 0$.

1. The target. $a \times c = -6$ and $b = -5$.
2. The pair. $-6$ and $1$: product $-6$, sum $-5$.
3. Split and group. $3x^2 - 6x + x - 2 = 3x(x - 2) + (x - 2) = (3x + 1)(x - 2)$.
4. Roots. $x = 2$ or $x = -\dfrac{1}{3}$. Check: $12 - 10 - 2 = 0$.

Practice with new numbers

A program to try

The program finds the split for the worked example and checks both roots.

Mistakes to avoid

• Deciding too early. Simplify before you call an equation quadratic.
• Using $c$ instead of $a \times c$ when you search for the split.
• Wrong sign in a root. $(x + 4) = 0$ gives $x = -4$.
• Dividing by $x$. It loses the root $x = 0$.
• Keeping an impossible root in a word problem.

Check yourself

1. Solve $x^2 - 8x + 15 = 0$.

2. Is $x(x + 1) = x^2 + 4$ a quadratic equation?

3. A rectangle is $2\ \text{m}$ longer than it is wide, and its area is $48\ \text{m}^2$. Find its width.

4. What are all the roots of $x^2 = 3x$?