Class 10 Maths · Chapter 4 · Topic 1 of 3
Simplify first: is it really quadratic?
Decide the degree only after simplifying the equation.
Ideas you’ll use
- Quadratic equation
- An equation ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
After cancellation, this is a straight line
p(x) (number)
x (number)
- Left side minus right side: 2x − 8 · Original teaching example; exact values are available below.
- 1. Original equation is true · (4, 0)
Assumptions: Lines join calculated sample points. Curves are visual approximations; use the formula and marked exact values for proof.
Read the plotted values
Left side minus right side: 2x − 8
| x (number) | p(x) (number) |
|---|---|
| 0 | -8 |
| 0.15 | -7.7 |
| 0.3 | -7.4 |
| 0.45 | -7.1 |
| 0.6 | -6.8 |
| 0.75 | -6.5 |
| 0.9 | -6.2 |
| 1.05 | -5.9 |
| 1.2 | -5.6 |
| 1.35 | -5.3 |
| 1.5 | -5 |
| 1.65 | -4.7 |
| 1.8 | -4.4 |
| 1.95 | -4.1 |
| 2.1 | -3.8 |
| 2.25 | -3.5 |
| 2.4 | -3.2 |
| 2.55 | -2.9 |
| 2.7 | -2.6 |
| 2.85 | -2.3 |
| 3 | -2 |
| 3.15 | -1.7 |
| 3.3 | -1.4 |
| 3.45 | -1.1 |
| 3.6 | -0.8 |
| 3.75 | -0.5 |
| 3.9 | -0.2 |
| 4.05 | 0.1 |
| 4.2 | 0.4 |
| 4.35 | 0.7 |
| 4.5 | 1 |
| 4.65 | 1.3 |
| 4.8 | 1.6 |
| 4.95 | 1.9 |
| 5.1 | 2.2 |
| 5.25 | 2.5 |
| 5.4 | 2.8 |
| 5.55 | 3.1 |
| 5.7 | 3.4 |
| 5.85 | 3.7 |
| 6 | 4 |
The difference is zero at 4. No square term survives, so this example is linear.
Expand both sides
Do not assume every equation containing x² is quadratic.
x(x+2)=x²+8In the picture: Compare the graph after simplification.
Read all the reasoning together
Expand both sides
Do not assume every equation containing x² is quadratic.
x(x+2)=x²+8Cancel matching terms
Subtract x² from both sides and move the remaining terms.
2x−8=0Read the coefficients
A quadratic has ax²+bx+c=0 with a≠0. Here a=0, so it is linear.
x=4Check the original
A simplified root must still fit the starting equation.
4(4+2)=4²+8=24
Decide the degree only after simplifying the equation.
Worked example: The square terms can disappear
x(x + 2) = x² + 8 ⇒ 2x − 8 = 0This is linear, because the x² terms cancel. It has one root, x = 4; the original equation contains squares but is not quadratic.
Read the supporting explanation
Expand brackets, combine like terms, and move all terms to one side. Read signed a, b and c only after doing this. A missing term has coefficient zero.
Go deeper: A parameter can change the degree
In kx² + 3x + 1 = 0, the equation is quadratic only if k ≠ 0. If k = 0, solve the remaining linear equation instead.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
Write 5x = 2x² − 3 in standard form and identify a, b, c.
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