Grade 10 Math | Let's Achieve Together

Before we get into factoring, you need to know the difference between two equation types you will see constantly in Grade 10.

A linear equation has x to the power of 1. Its graph is always a perfectly straight line. The slope never changes, which is why it stays straight.

A quadratic equation has x to the power of 2. Its graph curves into a U-shape called a parabola. The rate of change is not constant. The general form is y = ax² + bx + c.

Linear — y = 2x − 1

Quadratic — y = x² − 4

Key rule: See x²? Quadratic — graph curves. No x²? Linear — always straight.

The points where a parabola crosses the x-axis are called roots. Finding them is what factoring is all about.

Factoring means rewriting an expression as a product of simpler parts — the reverse of expanding. The very first thing you always try is pulling out the Greatest Common Factor (GCF): a number or variable that divides evenly into every single term.

The process: find the largest number and/or variable that divides into all terms, write it out front, then put the remaining terms in brackets.

Example 1
6x² + 9x
GCF = 3x (3 divides 6,9 and x divides x²,x)
= 3x(2x + 3)

Example 2
4x² − 8x + 12
GCF = 4 (4 divides 4,8,12 — no x in every term)
= 4(x² − 2x + 3)

Always check by expanding. Multiply the GCF back through — if you get the original, you factored correctly.

Common mistake: forgetting to divide every term inside the bracket. All terms must be divided by the GCF.

A trinomial is an expression with three terms like x² + 7x + 12. When there is no number in front of x², we use a simple and reliable method.

Find two numbers that multiply to c (the last number) and add to b (the middle number). List factor pairs of c, find the pair that sums to b, then write (x + p)(x + q).

Factor: x² + 7x + 12
Multiply to 12, add to 7
Pairs: (1,12),(2,6),(3,4) → 3×4=12 ✓ 3+4=7 ✓
= (x + 3)(x + 4)

Factor: x² − 5x + 6
Multiply to 6, add to −5
Both negative: (−2)(−3)=6, (−2)+(−3)=−5 ✓
= (x − 2)(x − 3)

Sign rules: c positive + b negative → both numbers negative. c negative → one positive, one negative.

This is a pattern worth memorizing — it lets you factor instantly with no trial and error. When you see two perfect squares being subtracted, the answer always follows the exact same structure.

The rule: a² − b² = (a + b)(a − b). There is no middle term — just two terms with a minus sign between them.

Pattern: a² − b² = (a + b)(a − b)

x² − 9 = x² − 3² = (x + 3)(x − 3)
x² − 25 = x² − 5² = (x + 5)(x − 5)
x² − 49 = x² − 7² = (x + 7)(x − 7)
4x² − 81 = (2x)² − 9² = (2x + 9)(2x − 9)

3-part checklist: First term a perfect square? Last term a perfect square? Minus sign between them? Yes to all three → difference of squares!

Only subtraction works. x² + 16 cannot be factored — a sum of squares does not factor.

Here is where everything connects. When you factor a quadratic and set it equal to zero, the solutions are called the roots — the exact x-values where the parabola crosses the x-axis.

Once you have the roots, the vertex sits exactly halfway between them. Find the vertex x-value by averaging the two roots. This chain — factor, roots, vertex — is the backbone of every Grade 10 quadratics question.

y = x² − 5x + 6, factored:
y = (x − 2)(x − 3)

Set y = 0, solve each bracket:
x − 2 = 0 → x = 2 (root)
x − 3 = 0 → x = 3 (root)

Axis of symmetry = average of roots:
x = (2 + 3) ÷ 2 = 2.5

The chain: Factor → roots → axis of symmetry → vertex. Every Grade 10 quadratics question uses part or all of this.