📘 Quadratic Equations – Class 10 (CBSE)
Quadratic equation – any equation that can be written in the form
[ ax^{2}+bx+c=0 \qquad (a\neq 0) ]
1. Simple Explanation
- A quadratic equation is a second‑degree polynomial equation.
- It has two roots (solutions) that can be real or complex.
- The graph of (y=ax^{2}+bx+c) is a parabola that opens upward ((a>0)) or downward ((a<0)).
2. Key Points
| Topic | What you need to know |
|---|---|
| Standard form | (ax^2+bx+c=0). Coefficients (a,b,c) are real numbers, (a\neq0). |
| Roots (solutions) | Values of (x) that satisfy the equation. |
| Discriminant ((\Delta)) | (\Delta = b^2-4ac). Determines nature of roots. |
| Nature of roots | • (\Delta>0): two distinct real roots • (\Delta=0): one real double root • (\Delta<0): two complex conjugate roots |
| Quadratic formula | [ |
| x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} | |
| ] | |
| Factoring method | Find two numbers (p,q) such that (p+q=b) and (pq=ac). |
| Completing the square | Re‑express (ax^2+bx+c) as (a(x-h)^2+k). |
| Graph | Vertex (\bigl(-\frac{b}{2a},,\frac{4ac-b^2}{4a}\bigr)). |
| Sum & Product of roots | If roots are (r_1,r_2): (; r_1+r_2=-\frac{b}{a}), (r_1r_2=\frac{c}{a}). |
| Special case | (x^2+px+q=0) (monic quadratic). |
| Applications | Projectile motion, area maximisation, etc. |
3. Worked Example
Solve
[
3x^2-12x+9=0
]
Step‑by‑Step
-
Identify coefficients
(a=3,; b=-12,; c=9). -
Compute discriminant
[ \Delta = b^2-4ac = (-12)^2-4(3)(9)=144-108=36 ] Since (\Delta>0), two distinct real roots. -
Apply quadratic formula
[ x = \frac{-b\pm\sqrt{\Delta}}{2a} = \frac{-(-12)\pm\sqrt{36}}{2\cdot 3} = \frac{12\pm 6}{6} ] -
Find the two roots
- (x_1 = \frac{12+6}{6}= \frac{18}{6}=3)
- (x_2 = \frac{12-6}{6}= \frac{6}{6}=1)
-
Check (optional)
Substitute (x=3) and (x=1) back into the original equation – both satisfy it.
Answer: (x=1) or (x=3).
4. Quick Revision Summary
- Standard form: (ax^2+bx+c=0).
- Discriminant: (\Delta=b^2-4ac).
- Nature of roots:
- (\Delta>0) → two real roots
- (\Delta=0) → one repeated real root
- (\Delta<0) → two complex conjugate roots
- Quadratic formula: (x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}).
- Factoring method: find (p,q) with (p+q=b) and (pq=ac).
- Completing the square: turn (ax^2+bx+c) into (a(x-h)^2+k).
- Sum & product of roots: (r_1+r_2=-\frac{b}{a}), (r_1r_2=\frac{c}{a}).
- Vertex of parabola: (\bigl(-\frac{b}{2a},,\frac{4ac-b^2}{4a}\bigr)).
Keep practising by solving equations using all three methods (formula, factoring, completing the square) to build confidence!