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Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

Squaring a Binomial: Don't Forget the Middle Term

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 2 min 18 s
  • British Columbia
  • BC10-MA-30

An area model splits a square with sides x + 4 into x², two 4x strips and 16, which shows that (x + 4)² = x² + 8x + 16 and that the common mistake x² + 16 leaves out the two strips; substituting x = 1 checks this (25, not 17, which is 8 short). It then shows (x + 5)(x − 5) = x² − 25, where the middle terms cancel because +5 and −5 are opposites, and ends on the recap "(x + 4)² = x² + 8x + 16: never forget the middle term".

Concept: Squaring a binomial: do not forget the middle term

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In one sentence

The question this lesson answers

Let us expand x plus 4, all squared. Expand means multiply out the brackets.

A binomial is an expression with two terms, here x and 4.

Transcript

What the lesson says

Let us expand x plus 4, all squared. Expand means multiply out the brackets.

A binomial is an expression with two terms, here x and 4.

A common mistake is to square each term on its own: x squared plus 16.

But squaring means multiplying by itself: x plus 4, times x plus 4.

Picture a square with sides x plus 4. Its area is x plus 4, squared.

Split each side into x and 4.

That makes four pieces: a big square, two strips and a small square.

The big square is x by x, or x squared. Each strip is x by 4, or 4x. The small square is 4 by 4, or 16.

x squared plus 16 counts only the two squares. It leaves out both strips.

Add all four pieces: x squared plus 4x plus 4x plus 16.

4x and 4x are like terms, with the same variable, x. Together they make 8x, the middle term.

Check with x equal to 1. x plus 4 is 5, and 5 squared is 25.

x squared plus 8x plus 16 gives 1 plus 8 plus 16, also 25.

But x squared plus 16 gives 1 plus 16, only 17. It is 8 short: the middle term, 8 times 1.

Now try x plus 5, times x minus 5. Multiply each term by each term in the other bracket.

The middle terms, minus 5x and plus 5x, cancel out, leaving x squared minus 25.

They cancel because plus 5 and minus 5 are opposites. In x plus 4, squared, they add to 8x.

So x plus 4, all squared, is x squared plus 8x plus 16. Never forget the middle term.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Algebra: multiplication of polynomial expressionsBC10-MA-30

Previous: Multiplying a binomial by a trinomialNext: Factoring Out the Greatest Common Factor

The outcomes and strands follow the British Columbia curriculum.

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