Lesson 3 of 4

Multiplying two brackets.

About 25 minutes · 5 practice questions

A product of two sums is a rectangle with four parts: where the middle term comes from, why the square of a sum is more than the sum of the squares, and what happens when a side has a minus.

Step 1 of 5 · Four parts

Step 1 of 5

Four parts

When both sides are sums, as in $(x + 3)(x + 2)$ ⁠, each side is split in two, and the rectangle is cut into four parts. Each term of one side is multiplied by each term of the other. Predict the result, then watch the parts fill in one by one.

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Step 2 of 5

The middle term

The two middle parts have the same colour, because both are x terms: they are like terms, and you collect them into one. Answer, then choose Split into parts under Look inside to check. Move x too: the parts change size, but the numbers in front of the letters stay the same.

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Step 3 of 5

The square of a sum

$(a + b)^2$ ⁠ means $(a + b)(a + b)$ ⁠: a square with side $a + b$ ⁠. Here the sides are made of two letters, a and b, each standing for a number. Predict how many parts the square has and what they say.

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Step 4 of 5

The square in numbers

From the parts, $(a + b)^2 = a^2 + 2ab + b^2$ ⁠, for every a and every b. Check it with numbers: here $a = 5$ ⁠ and $b = 3$ ⁠. Answer, then choose Split into parts under Look inside and compare the area with the sum of the parts.

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Step 5 of 5

The square of a difference

$x - 3$ ⁠ is also a side with two terms: $x$ ⁠ and $-3$ ⁠. A part with a minus is drawn pale, and its area is taken away. Watch the corner, where two minus signs meet. Predict, then see.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Predict, then look

The sides are $x + 3$ ⁠ and $x + 2$ ⁠. Predict: what is $(x + 3)(x + 2)$ ⁠ without brackets?

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Practice · Calculate

The sides are now $x + 4$ ⁠ and $x + 3$ ⁠. After collecting like terms, $(x + 4)(x + 3) = x^2 + kx + 12$ ⁠. What is $k$ ⁠?

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Practice · Predict, then look

Now both sides are $a + b$ ⁠, so this is a square: $(a + b)^2$ ⁠. Predict: what is it equal to?

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Practice · Calculate

$a = 5$ ⁠ and $b = 3$ ⁠. What is $(a + b)^2$ ⁠?

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Practice · Predict, then look

Both sides are $x - 3$ ⁠. A part with a minus is drawn pale: its area is taken away. Predict: what is $(x - 3)^2$ ⁠ equal to?