Quadratic Equations

Master the 'Sign Method' to solve quadratics in 5 seconds. Learn to compare X and Y without solving the full equation.

1. The Sign Method (5-Second Hack)

Predict the signs of the roots just by looking at the equation sign. This helps you compare X and Y instantly without calculating values.

Golden Rule: If Constant terms (

c

) in both equations are Negative, the answer is ALWAYS 'Relationship Cannot be Established' (CND).

Example:

Q: I.

x^2 - 7x + 12 = 0

II.

y^2 + 5y + 6 = 0

Solution: 1. Eq I is (-+), so roots are (+, +).
2. Eq II is (++), so roots are (-, -).
3. Positive > Negative.
Ans:

x > y

(No calculation needed!).

Find two numbers that multiply to $a \times c$ and add/subtract to $b$ . Then Divide by 'a' and Change Variable Sign.

Example:

2x^2 - 9x + 10 = 0

Solution: 1. Product

2 \times 10 = 20

. Sum

-9

.
2. Factors:

-4, -5

.
3. Change Signs:

+4, +5

.
4. Divide by

a=2

4/2, 5/2

.
Roots:

2, 2.5

Systematically compare both X roots with both Y roots.

Example:

x = 4, 3

and

y = 3, 2

Solution:

4 > 3, 4 > 2

(Pass).

3 = 3, 3 > 2

(Pass).
Ans:

x \ge y

Sometimes exams ask for $x$ and $y$ from linear sets like $2x + 3y = 12$ . Use Co-efficient Elimination.

Example:

Q: I.

2x + 3y = 13

II.

4x - y = 5

Solution: Multiply Eq II by 3

\to 12x - 3y = 15

.
Add with Eq I:

(2x+12x) + (3y-3y) = 13+15 \to 14x = 28 \to x=2

.
Put

x=2

in II:

8 - y = 5 \to y=3

.
Ans:

x < y