Quadratic Equations · Grade 9
How do you expand brackets and what are the special products?
Multiply every term in the first bracket by every term in the second, then collect like terms. Three formulas shortcut the common cases: the square of a sum, the square of a difference, and the difference of two squares. They are not extra rules but results of the same expansion, worth recognising on sight.
Learning objectives
- Multiply two brackets and collect the result into standard form
- Apply the formulas for the square of a sum, the square of a difference and the difference of squares
- Recognise a quadratic expression and name its three coefficients
Every term with every term
Multiplying two brackets means multiplying each term in the first by each term in the second. Two brackets of two terms give four products — always four, and that is the number to check against.
After multiplying, collect like terms. The result is a quadratic expression in standard form: a term in x squared, a term in x, and a constant.
Those three coefficients are what the rest of this topic works with. It is worth getting into the habit of saying them out loud, signs included, immediately after collecting.
Three formulas that save time
The square of a sum expands to the first squared, plus twice the product of the two, plus the second squared. The middle term is the one people drop.
The square of a difference is identical except that the middle term is negative. The difference of two squares is the elegant case: a sum times a difference of the same two terms cancels the middle term entirely and leaves only the difference between the squares.
Why it pays off
These formulas are not only shortcuts. They are also what lets you recognise, later, an expression that factorises at a glance — which is the first step in solving a quadratic equation in the lessons that follow.
Anyone who spots a difference of two squares saves themselves the entire quadratic formula. That is why it is worth knowing all three in both directions: from brackets to expanded form and back again.
Worked examples
Expand (x + 3)(x + 5)
- Multiply each with each: x times x, x times 5, 3 times x, 3 times 5
- This gives x squared, plus 5x, plus 3x, plus 15
- Collect the two x terms: 5x plus 3x is 8x
Answer: x squared plus 8x plus 15
Expand (x − 4) squared using a formula
- This is a squared difference
- First squared: x squared
- Twice the product, with a minus sign: minus 8x
- Second squared: 16
Answer: x squared minus 8x plus 16
Expand (x + 7)(x − 7)
- This is a sum times a difference of the same two terms — a difference of squares
- The middle terms cancel: plus 7x and minus 7x
- What remains is the first squared minus the second squared
Answer: x squared minus 49
Common mistakes
- Thinking a squared sum is a sum of squares
- The bracket squared includes the middle term, twice the product. Check it with numbers: two plus three, squared, is twenty-five, not four plus nine.
- Losing one of the four products
- Two brackets of two terms give exactly four products. A quick count before collecting catches the missing one, and this error disappears entirely once you count.
- Dropping a minus sign during multiplication
- Minus times minus gives plus, and minus times plus gives minus. The sign belongs to the term that follows it, so write them together before you start multiplying.
What to remember
- Every term with every term — four products.
- A squared bracket has a middle term: twice the product.
- A sum times a difference gives a difference of squares.
- Collect like terms and name all three coefficients.