Chapter 4 — Quadratic Equations: Quality Notes
1. What is a Quadratic Equation?
A quadratic equation in x has the standard form ax2+bx+c=0,a=0
where a,b,c are real numbers.
- The highest power of x must be 2 after simplifying the equation.
- An equation may not look quadratic initially; always simplify it first.
- If the x2 terms cancel completely, the equation is not quadratic.
Example: (x−2)2+1=2x−3
Simplifying gives x2−6x+8=0
so it is quadratic.
Key idea
Degree is determined after simplification, not from the appearance of the original equation.
2. Forming Quadratic Equations from Situations
Many real-life problems can be converted into quadratic equations by:
- Choosing an unknown, usually x.
- Expressing other quantities in terms of x.
- Using the given relationship.
- Rearranging everything to one side.
Example: Rectangle
If breadth =x and length is one more than twice the breadth, length=2x+1
For area 300m2, x(2x+1)=300
so 2×2+x−300=0
The important skill is translating words into algebra.
3. Roots of a Quadratic Equation
For ax2+bx+c=0
a number α is called a root/solution if substituting x=α makes the equation true: aα2+bα+c=0
The roots of the quadratic equation are the same as the zeroes of the corresponding quadratic polynomial.
A quadratic equation can have at most two roots.
4. Solving by Factorisation
If the quadratic can be expressed as two linear factors, (ax+b)(cx+d)=0
then use the zero-product property: (ax+b)=0or(cx+d)=0
and solve both linear equations.
Example
2×2−5x+3=0
Split the middle term: 2×2−2x−3x+3=0
Factor: 2x(x−1)−3(x−1)=0 (2x−3)(x−1)=0
Therefore, 2x−3=0orx−1=0
Hence, x=23,1
Factorisation strategy
For ax2+bx+c
look for two terms whose:
- product gives acx2
- sum gives bx
Then group and factor.
5. Choosing the Meaningful Root
Algebra may produce two roots, but a word problem may allow only one.
For example, if x represents a length, age, distance, number of objects, or breadth, a negative value may not make physical sense.
In the prayer-hall problem: (x−12)(2x+25)=0
gives x=12orx=−12.5
Since x represents breadth, only x=12
is meaningful.
Thus the dimensions are: breadth=12m,length=25m
Exam habit: Never stop after solving the equation. Check whether each root fits the original situation.
6. Quadratic Formula
For ax2+bx+c=0
the roots are x=2a−b±b2−4ac
This method is particularly useful when factorisation is difficult or inconvenient.
The expression b2−4ac
is extremely important.
It is called the discriminant.
7. Nature of Roots
Let D=b2−4ac
Then:
| Discriminant | Nature of roots |
|---|---|
| D>0 | Two distinct real roots |
| D=0 | Two equal real roots |
| D<0 | No real roots |
Why this matters
You can determine whether real solutions exist without actually solving the equation.
For example, 2×2−4x+3=0
has D=(−4)2−4(2)(3)=16−24=−8
Since D<0, there are no real roots
8. Equal Roots
When D=0
the quadratic has two equal real roots.
The repeated root is x=−2ab
So, if a question asks for the value of a parameter k that makes the roots equal, the fastest approach is usually: b2−4ac=0
and solve for k.
9. Quadratic Equations in Geometry and Daily Life
Quadratic equations are useful when a situation contains products of unknown quantities or squared quantities.
Common setups include:
Rectangle
Area=length×breadth
Consecutive integers
If the first integer is x, the next is x+1
so their product becomes x(x+1)
Speed–time–distance
Time=SpeedDistance
Changing the speed can produce a quadratic equation.
Geometry
Pythagoras’ theorem can lead to a2+b2=c2
and therefore to a quadratic equation when one unknown is involved.
10. A Useful Problem-Solving Framework
For word problems, use this sequence:
Understand → Choose x → Form expressions → Build equation → Simplify → Solve → Reject impossible roots → Answer with units
This prevents a common mistake: solving the algebra correctly but giving an answer that does not fit the original situation.
11. Must-Remember Formula Sheet
Standard form
ax2+bx+c=0,a=0
Quadratic formula
x=2a−b±b2−4ac
Discriminant
D=b2−4ac
Nature of roots
D>0D=0D<0⇒two distinct real roots⇒two equal real roots⇒no real roots
Equal root
x=−2ab
Factorisation principle
pq=0⇒p=0 or q=0
These are the core results highlighted in the chapter summary.
One-line chapter takeaway
A quadratic equation is an equation of degree 2; factorisation or the quadratic formula finds its roots, while the discriminant tells us what kind of real roots it has.