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Y320 · Quadratic Equations
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Middle school maths, year 3 · unit 3

Quadratic Equations

A flower bed 2 m longer than it is wide has an area of 15 m². How wide is it? As an equation that is x(x + 2) = 15, where x rises to a square. An equation like this can have two answers. We start by trying numbers one at a time.

Two students in a school garden measuring the length and width of a rectangular flower bed with a tape

A solution makes it 0

A board to arrange

Which x makes x² − 9 = 0 true? Drag number chips into the equation and check them one by one.

A product is 0 when one factor is

A board to adjust

How do you find the solutions of (x − 1)(x − 4) = 0? Change x with the handle and look for where the rectangle goes flat.

Solving by factorising

Joining board

Written as a product, the solutions show at once. Draw a line from each equation to its solutions.

Solving by square roots

A board to adjust

(x − 2)² = 4 can be solved without factorising. On the number line, spread out equally from the centre.

Completing the square

A board to adjust

What must be added to x² + 6x to make a perfect square? Fill the empty corner with unit tiles.

Slotting into the formula

A board to arrange

There is a formula that solves any quadratic. Drag a, b and c from the equation into its slots.

Try it yourself

Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.

What to take away from this unit

  1. An equation that becomes (a quadratic in x) = 0 when every term is moved to the left is a quadratic equation. The x that makes it true is a solution, and there are usually two.
  2. If AB = 0 then A = 0 or B = 0. So factorising into a product solves it: (x − 2)(x − 3) = 0 gives x = 2 or x = 3.
  3. When the two solutions coincide, as in (x − 3)² = 0, there is one solution, a repeated root.
  4. If x² = q then x = ±√q, and if (x − p)² = q then x = p ± √q. Keep both the positive and the negative.
  5. Add the square of half of b to x² + bx and it becomes a perfect square. With this, any quadratic can be put in the form (x − p)² = q.
  6. The quadratic formula: the solutions of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a. It works even when factorising does not.

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