Quadratic Quest
Hunt down the roots of a quadratic: factorise it into two brackets, use the zero-product rule to read off the solutions, and see where those roots live on the parabola.
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Quadratic Quest 🗺️
A **quadratic** contains an **x²** term. Solving one means finding its **roots** — the values of x that make it equal zero, and there can be **two** of them. The main weapon is **factorising** (from Expand & Conquer) plus one simple rule about zero. Let the quest begin.
What is a quadratic? 🧮
A quadratic equation looks like **x² + bx + c = 0** — the highest power of x is 2. Its graph is a **parabola** (a U-shape), and where that curve crosses the x-axis gives the **roots**. Because a parabola can cross the axis twice, a quadratic usually has two solutions.
Evaluate the expression
An interactive activity.
The zero-product rule 🟰
Here is the key idea. If two things multiply to give **zero**, then **at least one of them must be zero**: **If P × Q = 0, then P = 0 or Q = 0.** So once a quadratic is written as two brackets multiplied together and set equal to 0, you can solve each bracket separately.
Solve by factorising ➗
To solve **x² + 5x + 6 = 0**: 1. **Factorise:** two numbers multiply to 6 and add to 5 → 2 and 3, so (x + 2)(x + 3) = 0. 2. **Set each bracket to 0:** x + 2 = 0 or x + 3 = 0. 3. **Solve each:** x = **−2** or x = **−3**.\n\nNotice the roots are the **negatives** of the numbers in the brackets.
Order the method
An interactive activity.
Mind the sign
One bracket of a factorised quadratic is (x + 2) = 0. What value of x does this give?
- x = −2
- x = 2
- x = −½
- x = 0
The other root
An interactive activity.
A second example 🔁
Signs work the same way when the numbers are negative. **x² − 7x + 12 = 0**: two numbers multiply to **+12** and add to **−7** → −3 and −4. So (x − 3)(x − 4) = 0, giving x = **3** or x = **4**. (A minus in the bracket makes a positive root.)
Solve it
An interactive activity.
Match each quadratic to its roots
- x² + 5x + 6 = 0
- x² − 7x + 12 = 0
- x² + x − 6 = 0
- x² − 9 = 0
- x = −2 and −3
- x = 3 and 4
- x = 2 and −3
- x = 3 and −3
A special case ◻️
When there is **no x term** and a subtraction, it is the **difference of two squares**. **x² − 9 = 0** has no middle term: x² − 9 = (x − 3)(x + 3) = 0, so x = **3** or x = **−3**. The two roots are simply **+ and −** the square root of the number.
Difference of two squares
An interactive activity.
Roots on the graph 📉
The roots you calculate are exactly where the **parabola crosses the x-axis** (where y = 0). For y = (x − 2)(x − 4), the curve crosses the x-axis at x = 2 and x = 4 — so those crossing points are (2, 0) and (4, 0). Solving and graphing tell the same story.
Plot the root
An interactive activity.
The quest rules
A quadratic equation can have up to _____ roots. To solve it by factorising, write it as two brackets multiplied together equal to zero, then set each bracket equal to _____. On a graph, the roots are where the parabola crosses the _____-axis.