Connected ideas

Quadratics and areas of shapes

Change dimensions. Connect rectangles, squares and triangles with quadratic expressions.

Predict, test and explain

Step 1 of 3

Predict

A square has side x + 3. What is its expanded area?

A=(x+3)(x+4)

Scroll the diagram sideways to see every label.

x=2

A=30\;\text{square units}

Each side contains x. The four regions give x squared, two linear terms and a constant. Add their areas to expand the product.

Check the working

Quantity	Value
Width / base	$x+4=6$
Perpendicular height	$x+3=5$
Region 1	$x^2=4$
Region 2	$3x=6$
Region 3	$4x=8$
Region 4	$12=12$
Area expression	$A=(x+3)(x+4)=x^2 + 7x + 12$
Area at x = 2	$A=30\;\text{square units}$

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

A square has side (x + 3) units. Expand its area and find the area when x = 2.

$A=(x+3)^2$

Hint

Multiply the side by itself. Include both products involving one x and one three.

Reveal answer

The two linear products combine to give six x. When x is two, the side is five, so the area is twenty-five square units.

$A=x^2+6x+9,\quad A(2)=25$

Build an explanation

Hint 1

Split a rectangle with sides x + a and x + b at length x on each side. Both dimensions change with x.

Hint 2

The four regions have areas x², ax, bx and ab. Add them to get x² + (a + b)x + ab.

Hint 3

A square uses equal sides, so a = b. A triangle with the same base and perpendicular height has half the rectangle’s area.

Worked example

This example uses fixed values, separate from the diagram controls.

A rectangle has sides (x + 3) and (x + 4) units. Expand its area and find the area when x = 2.

Multiply the two side lengths: A = (x + 3)(x + 4).
The four products are x², 3x, 4x and 12. Collect like terms: A = x² + 7x + 12.
At x = 2, the dimensions are 5 by 6, giving an area of 30 square units.
The expanded form gives 4 + 14 + 12 = 30. A triangle with this base and perpendicular height has area 15 square units.

Answer: A = x² + 7x + 12; at x = 2 the area is 30 square units.

Connect this idea

Watch out: Area uses multiplication, not addition of side lengths. (x + 3)² = x² + 6x + 9, including both linear regions. All lengths in this model are positive.