Quadratics Lesson
Hello. I am going to teach you everything you need to know about quadratics. I will assume you know nothing — and by the end, you will be solving them confidently. Let us begin.
What is a Quadratic?
A quadratic is a type of equation where the highest power of x is x² (x squared). The word comes from the Latin "quadratus" meaning square — because x² gives you a square-shaped area.
Every quadratic can be written in the form:
Here, a, b, and c are just numbers (constants). The only rule is that a cannot be zero — if a = 0, then x² disappears and we do not have a quadratic anymore.
Examples of Quadratics
✅ x² + 5x + 6 = 0 — this is a quadratic (a = 1, b = 5, c = 6)
✅ 2x² − 3x + 1 = 0 — this is a quadratic (a = 2, b = −3, c = 1)
✅ x² − 9 = 0 — this is a quadratic (a = 1, b = 0, c = −9)
❌ x + 5 = 0 — this is not a quadratic (no x² term)
❌ x³ + x² = 0 — not a quadratic (x³ is higher than x²)
Teacher's Tip: When you see x², think "quadratic." The x² term is what makes it special. Everything else is just detail.
Solving — Method 1: Factorising
"Solving" a quadratic means finding the value(s) of x that make the equation true. The simplest way to do this (when it works) is factorising.
How Factorising Works
We want to rewrite ax² + bx + c as a product of two brackets. For the simplest case where a = 1:
We need two numbers that multiply to give c (which is 6) and add to give b (which is 5).
Think: what two numbers multiply to 6? → (1 and 6), (2 and 3), (−1 and −6), (−2 and −3).
Which pair adds to 5? → 2 + 3 = 5. Yes!
So: x² + 5x + 6 = (x + 2)(x + 3)
Therefore: (x + 2)(x + 3) = 0
If two things multiply to zero, one of them must be zero.
Either: x + 2 = 0 → x = −2
Or: x + 3 = 0 → x = −3
Checking Your Answer
Always check by substituting back into the original equation:
If x = −2: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0 ✓
If x = −3: (−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0 ✓
Both work. We have solved it correctly.
Two numbers that multiply to 10 and add to −7.
Think: (−5) × (−2) = 10 and (−5) + (−2) = −7.
(x − 5)(x − 2) = 0
x − 5 = 0 → x = 5
x − 2 = 0 → x = 2
Teacher's Tip: When factorising, always ask yourself: "What two numbers multiply to make c and add to make b?" Practise this question in your head and it becomes automatic.
Special Case: Difference of Two Squares
When the quadratic looks like x² − k², it factorises as (x − k)(x + k):
x² − 16 = (x − 4)(x + 4) = 0
x = 4 or x = −4
Method 2: Completing the Square
Not every quadratic can be factorised easily. Completing the square is a method that always works. It turns x² + bx into a perfect square.
In words: take half of b, put it inside the bracket, then subtract its square.
Step 1: Focus on x² + 6x. Half of 6 is 3.
x² + 6x = (x + 3)² − 9
Step 2: Put it back: (x + 3)² − 9 + 5 = 0
Step 3: Simplify: (x + 3)² − 4 = 0
Step 4: (x + 3)² = 4
Step 5: x + 3 = ±2
x = −3 + 2 = −1 or x = −3 − 2 = −5
Teacher's Tip: Completing the square is like "wrapping" the x² and x terms into a perfect square. The formula (x + p)² − p² is your best friend here.
Method 3: The Quadratic Formula
This is the most powerful method. It works for every single quadratic, no matter what. You just memorise one formula and plug in the numbers.
Given ax² + bx + c = 0, you identify a, b, and c, then substitute them into this formula. That is all.
Here a = 2, b = 5, c = −3.
Step 1: Plug into the formula:
x = (−5 ± √(5² − 4×2×(−3))) / (2×2)
Step 2: Simplify inside the square root:
= (−5 ± √(25 + 24)) / 4
= (−5 ± √49) / 4
= (−5 ± 7) / 4
Step 3: Two solutions:
x = (−5 + 7) / 4 = 2/4 = 0.5
x = (−5 − 7) / 4 = −12/4 = −3
Teacher's Tip: Write the formula down every time until you know it by heart. A common mistake is forgetting the ± sign — it gives you two answers, not one!
The Discriminant
Inside the quadratic formula, the part under the square root — b² − 4ac — has a special name: the discriminant. It tells you how many solutions a quadratic has without solving it.
The rule is simple:
🔵 If Δ > 0 → two different real solutions (the graph crosses the x-axis twice)
🟢 If Δ = 0 → one repeated real solution (the graph just touches the x-axis)
🔴 If Δ < 0 → no real solutions (the graph never touches the x-axis)
a = 1, b = 2, c = 5
Δ = 2² − 4×1×5 = 4 − 20 = −16
Since Δ = −16 < 0, there are no real solutions.
Graphs of Quadratics
A quadratic graph is called a parabola. It is a smooth U-shaped curve. The sign of a tells you which way it points:
✅ If a > 0 (positive), the parabola smiles — U shape (minimum at the bottom)
✅ If a < 0 (negative), the parabola frowns — ∩ shape (maximum at the top)
The graph below lets you play with a, b, and c to see how they change the shape. Drag the sliders and watch what happens.
↑ Back to menuy = x²
Practice — Step by Step
Solve each quadratic one step at a time. Type your answer for each step and click Check Step. You must get each step correct before moving to the next.
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- Sum and product rule: For x² + bx + c = 0, the two numbers that factorise it always have sum = −b and product = c. Actually — sum = −b only when the coefficient of x² is 1 and you've moved everything to one side.
- Checking factorisation: Expand (x + p)(x + q) in your head: it is x² + (p+q)x + pq. The middle coefficient is the sum, the constant is the product.
- Completing the square shortcut: x² + bx = (x + b/2)² − (b/2)². Half it, square it, subtract it. Never forget this pattern.
- Quadratic formula memory trick: Sing it to "Pop Goes the Weasel": "x equals negative b, plus or minus the square root, of b-squared minus four a c, all over two a".
- Discriminant at a glance: Δ = b² − 4ac. If Δ is a perfect square, the quadratic factorises nicely over integers. If not, use the formula.
- Standard form first: Always rewrite the equation as ax² + bx + c = 0 before doing anything else. This means moving all terms to one side.
- a ≠ 0: If a = 0, it is not a quadratic. The x² term is what makes it a quadratic — without it you just have a linear equation.
- Always check your answers: Substitute each solution back into the original equation. If it works, you are correct. This catches most mistakes.
- Three methods = three tools: Factorising (fastest, but not always possible), Completing the square (always works, good for finding turning points), Quadratic formula (always works, best when numbers are ugly). Choose the right tool.
- Graph connection: The solutions of ax² + bx + c = 0 are the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis. No crossing → no real solutions.
- Forgetting the ± sign: The quadratic formula gives two answers. If you only write one, you lost half the marks. Always include both.
- Wrong sign when factorising: (x − 2)(x + 3) gives x² + x − 6, not x² − x − 6. Check your signs by expanding in your head before writing the answer.
- Forgetting to set = 0: If you have x² + 5x = −6, you must first rewrite it as x² + 5x + 6 = 0 before factorising or using the formula.
- Losing the negative sign: In the formula, −b means the opposite of b. If b = −3, then −b = 3. A common error is writing −3 instead of 3.
- Misreading the discriminant: Δ = b² − 4ac. It is b², not (b)² with the sign included. b = −3 gives b² = 9, not −9.
The Golden Rule of Quadratics
When in doubt, use the formula. It always works. Factorising is faster but the formula never fails. Memorise it, own it, love it.
You Have Completed the Quadratics Lesson
Here is what you have learned:
- ✅ What a quadratic is and its standard form ax² + bx + c = 0
- ✅ Solving by factorising (find two numbers that multiply to c and add to b)
- ✅ Completing the square: (x + b/2)² − (b/2)²
- ✅ The quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
- ✅ The discriminant b² − 4ac tells you how many solutions exist
- ✅ Quadratic graphs are parabolas (U or ∩ shaped)
Well done. You now know quadratics. This is a major step — everything in Pure Mathematics builds on this.