8.1 Quadratic Equations & Root Comparison

8.1 Quadratic Equations & Root Comparison

In the Quantitative Aptitude section of the State Bank of India (SBI) Junior Associate (Customer Support & Sales) examination, quadratic equations—frequently termed Quadratic Inequalities or Root Comparison—represent one of the highest-yield topics. In the Preliminary Examination, a dedicated set of 5 questions appears regularly in Numerical Ability (comprising 14.3% of the 35-mark section). In the Main Examination, quadratic models reappear with higher computational complexity, featuring large composite coefficients, fractional terms, or radical roots.

The testing objective in SBI Clerk is not merely deriving roots through the classical quadratic formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$), which is far too slow for a 20-minute sectional window. Instead, top-scoring candidates employ the Universal Sign Method, Coefficient Cross-Multiplication, and Direct Sign Inspection to crack all 5 questions within 90 to 120 seconds.

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Standard Quadratic Architecture & The Universal Sign Method

A quadratic equation is defined in its canonical algebraic form as:

$ax^2 + bx + c = 0 \quad (a \neq 0)$

where $a$, $b$, and $c$ are real coefficients. By Vieta's formulas, the sum of the roots $(\alpha + \beta) = -\frac{b}{a}$ and the product of the roots $(\alpha \beta) = \frac{c}{a}$. The algebraic signs of coefficients $b$ and $c$ directly govern the polarity of the two roots $(\alpha, \beta)$.

The Two-Step Sign Inversion Algorithm

Rather than memorizing arbitrary rules, understand the two-step mechanical sign transformation:

1. First Root Sign: The sign of the linear coefficient $b$ is always reversed ($+ \rightarrow -$ and $- \rightarrow +$).
2. Second Root Sign: The sign of the constant term $c$ dictates the second root: if $c$ is positive ($+$), the second root takes the same sign as the first root; if $c$ is negative ($-$), the second root takes the opposite sign of the first root.

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Instant Inspection Shortcuts: Zero-Calculation Traps

In many quadratic sets, one or two pairs can be solved instantly without writing a single calculation.

Rule 1: The "Both Constant Terms Negative" Rule (Instant CND)

[!IMPORTANT] The Most Powerful SBI Shortcut: If the constant term $c$ in both quadratic equations is negative ($c_1 < 0$ and $c_2 < 0$), the mathematical relationship between the variables can never be established.

Official Exam Option: Relationship cannot be established (or $x = y$).

Mathematical Proof:

• When $c_1 < 0$, Equation I produces one positive root and one negative root: $x_1 > 0$ and $x_2 < 0$.
• When $c_2 < 0$, Equation II produces one positive root and one negative root: $y_1 > 0$ and $y_2 < 0$.
• When comparing $x_1$ ($+$) with $y_2$ ($-$), we obtain $x_1 > y_2$.
• When comparing $x_2$ ($-$) with $y_1$ ($+$), we obtain $x_2 < y_1$.
• Because both a greater-than ($>$) and a less-than ($<$) inequality emerge simultaneously across valid root pairings, a consistent single-direction relationship is impossible. The answer is unconditionally Relationship Cannot Be Established (CND).

Rule 2: Pure Sign Segregation ($x > y$ or $x < y$)

If one equation exhibits the sign pattern $(-, +)$, its roots are both strictly positive ($+, +$). If the competing equation exhibits the sign pattern $(+, +)$, its roots are both strictly negative ($-, -$).

• Since every positive real number is strictly greater than every negative real number, the variable with positive roots is unconditionally greater than the variable with negative roots.
• Example: If $x^2 - 14x + 45 = 0$ (roots $+9, +5$) and $y^2 + 17y + 72 = 0$ (roots $-8, -9$), then $x > y$ holds by pure inspection in under 3 seconds.

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Factoring Shortcuts for Large Coefficients ($a \times c$)

When leading coefficients exceed 1 ($a > 1$), the candidate must find two factors of the composite product $P = a \times c$ whose algebraic sum or difference equals $b$.

Prime Decomposition Splitting Method

When $a \times c$ is a large composite number, never attempt random trial-and-error guessing. Systematically decompose the product into small prime building blocks:

Example Equation: $12x^2 - 23x + 10 = 0$

1. Compute the product: $P = a \times c = 12 \times 10 = 120$.
2. Identify target sum: The sign pattern is $(-, +)$, requiring two factors whose sum is $+23$ and whose signs will both invert to positive ($+, +$).
3. Prime decompose $120$: $120 = 2^3 \times 3 \times 5 = 8 \times 15$.
4. Check sum: $8 + 15 = 23$. The factorization is exact!
5. Raw roots: $+15$ and $+8$.
6. True roots: Divide raw roots by $a = 12$: $x_1 = +\frac{15}{12} = +\frac{5}{4} = +1.25, \quad x_2 = +\frac{8}{12} = +\frac{2}{3} \approx +0.667$

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Cross-Multiplication Balancing: Eliminating Fraction Division

[!WARNING] The Fraction Division Trap: Dividing large raw roots by asymmetric coefficients ($a_1 = 6$ vs $a_2 = 10$) creates complex fractions (e.g., $15/6$ vs $21/10$) or recurring decimals, leading to computational fatigue and rounding mistakes.

To bypass fraction division entirely, use Opposite Coefficient Cross-Multiplication:

• Suppose Equation I has leading coefficient $a_1$ and raw roots $X_1, X_2$ (where true roots are $x_1 = X_1 / a_1, x_2 = X_2 / a_1$).
• Suppose Equation II has leading coefficient $a_2$ and raw roots $Y_1, Y_2$ (where true roots are $y_1 = Y_1 / a_2, y_2 = Y_2 / a_2$).
• Since $a_1 > 0$ and $a_2 > 0$, multiplying all roots by the common positive quantity $(a_1 \times a_2)$ preserves the exact inequality order: $x'_i = a_2 \times X_i \quad \text{and} \quad y'_j = a_1 \times Y_j$
• Comparing scaled integer roots $x'$ against $y'$ yields the exact mathematical comparison without performing any decimal divisions.

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The 4-Way Root Comparison Matrix

Comparing two roots of $x$ against two roots of $y$ requires testing all four possible pairs:

The Golden Decision Rules

1. Strict Superiority ($x > y$): All 4 comparisons yield $>$.
2. Strict Inferiority ($x < y$): All 4 comparisons yield $<$.
3. Weak Superiority ($x \ge y$): Comparisons yield a combination of $>$ and $=$, with no $<$ present.
4. Weak Inferiority ($x \le y$): Comparisons yield a combination of $<$ and $=$, with no $>$ present.
5. Identity ($x = y$): All 4 comparisons yield $=$.
6. Contradiction / Overlap (CND): Any comparison contains at least one $>$ AND at least one $<$. The relationship cannot be established.

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Step-by-Step Worked Examination Problems

Problem 1: Asymmetric Leading Coefficients with Cross-Multiplication

Equations: $\text{I. } 6x^2 - 19x + 15 = 0$ $\text{II. } 10y^2 - 29y + 21 = 0$

Step 1: Solve Equation I ($6x^2 - 19x + 15 = 0$):

• Product $P_1 = 6 \times 15 = 90$.
• Signs: $(-, +) \implies (+, +)$. Factors of 90 adding to 19: $10 \times 9 = 90$, $10 + 9 = 19$.
• Raw roots of $x$: $X_1 = +10, X_2 = +9$.
• Leading coefficient $a_1 = 6$.

Step 2: Solve Equation II ($10y^2 - 29y + 21 = 0$):

• Product $P_2 = 10 \times 21 = 210$.
• Signs: $(-, +) \implies (+, +)$. Factors of 210 adding to 29: $210 = 10 \times 21 = (2 \times 5) \times (3 \times 7) = 15 \times 14$. Note: $15 + 14 = 29$.
• Raw roots of $y$: $Y_1 = +15, Y_2 = +14$.
• Leading coefficient $a_2 = 10$.

Step 3: Apply Cross-Multiplication Balancing:

• Multiply raw roots of $x$ by $a_2 = 10$: $x'_1 = 10 \times 10 = 100, \quad x'_2 = 9 \times 10 = 90$
• Multiply raw roots of $y$ by $a_1 = 6$: $y'_1 = 15 \times 6 = 90, \quad y'_2 = 14 \times 6 = 84$

Step 4: Execute 4-Way Comparison:

1. $x'_1$ (100) vs $y'_1$ (90) $\implies 100 > 90$ ($x > y$)
2. $x'_1$ (100) vs $y'_2$ (84) $\implies 100 > 84$ ($x > y$)
3. $x'_2$ (90) vs $y'_1$ (90) $\implies 90 = 90$ ($x = y$)
4. $x'_2$ (90) vs $y'_2$ (84) $\implies 90 > 84$ ($x > y$)

Verdict: Comparisons reveal $>$ and $=$, with no $<$ relationship. Therefore, $x \ge y$.

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Quantity I vs. Quantity II Comparisons

Mains papers can present two quantities defined by short word problems instead of two equations. The answer options mirror root comparison: Quantity I > Quantity II, Quantity I < Quantity II, Quantity I ≥ Quantity II, Quantity I ≤ Quantity II, or Quantity I = Quantity II / relationship cannot be established.

Method

1. Evaluate each quantity completely and separately.
2. If a quantity has more than one possible value, compare every value with the other quantity, exactly like the 4-way root matrix.
3. Choose the single relation that holds for every possible pairing.

Worked Examples

[!WARNING] Hidden second value: Square roots of an unknown ($x^2 = k$), quadratic word problems and absolute values often hide a negative value. Check for it before choosing a strict relation.