7.2 Quadratic Equations, Nature of Roots, Discriminant Analysis & Maximum/Minimum Values

Standard Form & The Quadratic Formula (Sridharacharya's Rule)

A second-degree polynomial equation in a single variable $x$ is termed a quadratic equation. Its standard form is:

$$ax^2 + bx + c = 0, \quad \text{where } a, b, c \in \mathbb{R} \text{ and } a \neq 0$$

Derivation by Completing the Square

To understand the structural origin of the quadratic formula:

1. Divide through by the leading coefficient $a$:

$$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$

1. Transpose the constant term to the right side:

$$x^2 + \frac{b}{a}x = -\frac{c}{a}$$

1. Complete the square on the left side by adding $\left( \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2}$ to both sides:

$$x^2 + 2 \left( \frac{b}{2a} \right) x + \frac{b^2}{4a^2} = \frac{b^2}{4a^2} - \frac{c}{a} = \frac{b^2 - 4ac}{4a^2}$$

1. Factor the left side as a perfect square binomial:

$$\left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}$$

1. Take the square root of both sides:

$$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$

1. Isolate $x$ to yield the classical Sridharacharya Quadratic Formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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Vieta's Relations: Roots and Coefficients

Let $\alpha$ and $\beta$ be the two roots of the quadratic equation $ax^2 + bx + c = 0$:

$$\alpha = \frac{-b + \sqrt{b^2 - 4ac}}{2a}, \quad \beta = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$

The Fundamental Relations

• Sum of Roots ($\alpha + \beta$):

$$\alpha + \beta = \frac{-b + \sqrt{D}}{2a} + \frac{-b - \sqrt{D}}{2a} = \frac{-2b}{2a} = -\frac{b}{a}$$

• Product of Roots ($\alpha \beta$):

$$\alpha \beta = \left( \frac{-b + \sqrt{D}}{2a} \right) \left( \frac{-b - \sqrt{D}}{2a} \right) = \frac{(-b)^2 - (\sqrt{D})^2}{4a^2} = \frac{b^2 - (b^2 - 4ac)}{4a^2} = \frac{4ac}{4a^2} = \frac{c}{a}$$

Reconstructing the Quadratic Equation

Any quadratic equation can be directly assembled from its roots $\alpha$ and $\beta$:

$$x^2 - (\alpha + \beta)x + \alpha \beta = 0 \iff x^2 - (\text{Sum of Roots})x + (\text{Product of Roots}) = 0$$

+-----------------------------------------------------------------------------------------+
|                         SYMMETRIC FUNCTIONS OF ROOTS DIRECTORY                          |
+-----------------------------------------------------------------------------------------+
| Sum of Squares:              alpha^2 + beta^2 = (alpha + beta)^2 - 2*alpha*beta         |
|                                               = (b^2 - 2ac) / a^2                       |
| Difference of Roots:         |alpha - beta|   = sqrt((alpha + beta)^2 - 4*alpha*beta)   |
|                                               = sqrt(D) / |a|                           |
| Sum of Cubes:                alpha^3 + beta^3 = (alpha + beta)^3 - 3*alpha*beta(alpha + beta)  |
|                                               = -b(b^2 - 3ac) / a^3                     |
| Reciprocal Sum:              1/alpha + 1/beta = (alpha + beta) / (alpha*beta) = -b / c  |
| Reciprocal Square Sum:       1/alpha^2 + 1/beta^2 = (alpha^2 + beta^2) / (alpha*beta)^2 |
| Ratio Sum:                   alpha/beta + beta/alpha = (alpha^2 + beta^2) / (alpha*beta)|
|                                                      = (b^2 - 2ac) / (ac)               |
+-----------------------------------------------------------------------------------------+

For the absolute real-root difference, require $D\ge0$. Reciprocal-root expressions require $c\ne0$, so neither root is zero. Multiplying roots by $k$ via substitution requires $k\ne0$.

Useful Root Transformation Shortcuts

When a question requests an equation with transformed roots, direct polynomial substitution is faster than finding roots:

• Reciprocal Roots ($\frac{1}{\alpha}, \frac{1}{\beta}$): Replace $x$ with $\frac{1}{x} \implies cx^2 + bx + a = 0$ (swap $a$ and $c$).
• Roots with Opposite Signs ($-\alpha, -\beta$): Replace $x$ with $-x \implies ax^2 - bx + c = 0$ (negate $b$).
• Roots Multiplied by Constant ($k\alpha, k\beta$): Replace $x$ with $\frac{x}{k} \implies ax^2 + kbx + k^2c = 0$.

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Discriminant Analysis & The Nature of Roots

The quantity under the square root in Sridharacharya's formula is termed the Discriminant ($D$):

$$D = b^2 - 4ac$$

The algebraic value of $D$, relative to zero and the field of rational numbers, completely determines the nature and geometric orientation of the roots.

+-----------------------------------------------------------------------------------------+
|                              DISCRIMINANT DECISION TREE                                 |
+-----------------------------------------------------------------------------------------+
|                                    D = b^2 - 4ac                                        |
|                                          │                                              |
|             ┌────────────────────────────┼────────────────────────────┐                 |
|             ▼                            ▼                            ▼                 |
|          D > 0                         D = 0                        D < 0               |
|      Real & Distinct                Real & Equal                  Imaginary             |
|             │                     (Coincident Roots)         (Complex Conjugates)       |
|      ┌──────┴──────┐                   x = -b/(2a)                 p +- i*q             |
|      ▼             ▼               (Tangent to X-axis)      (No X-axis intersection)    |
|  D is Perfect   D is NOT                                                                |
|    Square     Perfect Square                                                            |
| Rational Roots  Irrational Conjugates                                                   |
|   p, q in Q        p +- sqrt(q)                                                         |
+-----------------------------------------------------------------------------------------+

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Graphical Properties & Extreme Values of Quadratic Functions

The graph of any quadratic function $f(x) = ax^2 + bx + c$ is a symmetric vertical parabola.

Vertex Coordinates & Axis of Symmetry

By completing the square on the function:

$$f(x) = a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a} \right) = a \left( x + \frac{b}{2a} \right)^2 - \frac{D}{4a}$$

• The Axis of Symmetry is the vertical line $x = -\frac{b}{2a}$.
• The Vertex of the parabola is located at the point:

$$\text{Vertex } (V) = \left( -\frac{b}{2a}, -\frac{D}{4a} \right) = \left( -\frac{b}{2a}, \frac{4ac - b^2}{4a} \right)$$

   a > 0 (Opens Upwards - Minimum)              a < 0 (Opens Downwards - Maximum)
               y                                            y
               |       /                                    |      Vertex (-b/(2a), -D/(4a))
               |      /                                     |         .---.
        \     |     /                                      |        /     \
         \    |    /                                 ------|-------/-------\-------> x
          \   |   /                                        |      /         \
           '--+--'  Vertex (-b/(2a), -D/(4a))              |     /           \
        ------+------------------> x                       |    /             \
              |                                            |

Maximum and Minimum Rules

1. Case 1: Leading Coefficient $a > 0$ (Parabola opens upwards):

   • Because $\left( x + \frac{b}{2a} \right)^2 \ge 0$, the minimum value occurs when that squared term vanishes.
   • Absolute Minimum Value: $f_{\min} = -\frac{D}{4a} = \frac{4ac - b^2}{4a}$, occurring at $x = -\frac{b}{2a}$.
   • Maximum Value: No finite maximum over all real $x$; the function is unbounded above.
   • Range: $\left[ -\frac{D}{4a}, \infty \right)$.

2. Case 2: Leading Coefficient $a < 0$ (Parabola opens downwards):

   • Because $a < 0$, $a \left( x + \frac{b}{2a} \right)^2 \le 0$. The maximum value occurs when the negative term is zero.
   • Absolute Maximum Value: $f_{\max} = -\frac{D}{4a} = \frac{4ac - b^2}{4a}$, occurring at $x = -\frac{b}{2a}$.
   • Minimum Value: No finite minimum over all real $x$; the function is unbounded below.
   • Range: $\left( -\infty, -\frac{D}{4a} \right]$.

Definiteness of Quadratic Forms

• Strictly Positive Definite: $ax^2 + bx + c > 0$ for all real $x$ if and only if $a > 0$ and $D < 0$.
• Strictly Negative Definite: $ax^2 + bx + c < 0$ for all real $x$ if and only if $a < 0$ and $D < 0$.

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Worked Examples

Worked Example 1: Constructing Quadratic Equations with Transformed Roots

Problem: If $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^2 - 7x + 4 = 0$, construct the quadratic equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.

• Step 1: Compute the basic root relations via Vieta's formulas:

$$\alpha + \beta = -\frac{-7}{2} = \frac{7}{2}$$ $$\alpha \beta = \frac{4}{2} = 2$$

• Step 2: Find the sum of the new roots ($S$):

$$S = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{(\alpha + \beta)^2 - 2\alpha \beta}{\alpha \beta}$$

Substitute the computed values:

$$S = \frac{\left(\frac{7}{2}\right)^2 - 2(2)}{2} = \frac{\frac{49}{4} - 4}{2} = \frac{\frac{49 - 16}{4}}{2} = \frac{\frac{33}{4}}{2} = \frac{33}{8}$$

• Step 3: Find the product of the new roots ($P$):

$$P = \left(\frac{\alpha}{\beta}\right) \left(\frac{\beta}{\alpha}\right) = 1$$

• Step 4: Form the quadratic equation:

$$x^2 - Sx + P = 0 \implies x^2 - \frac{33}{8}x + 1 = 0$$

Multiply through by 8 to express in integer coefficients:

$$8x^2 - 33x + 8 = 0$$

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Worked Example 2: Condition for Equal Roots

Problem: Determine the value of $m$ for which the quadratic equation $(m - 2)x^2 - (2m - 1)x + (m + 2) = 0$ has real and equal roots.

• Step 1: Identify coefficients and state the condition: Here $a = m - 2$, $b = -(2m - 1)$, and $c = m + 2$. For real and equal roots, the discriminant must vanish: $D = b^2 - 4ac = 0$.
• Step 2: Set up and expand the discriminant equation:

$$[-(2m - 1)]^2 - 4(m - 2)(m + 2) = 0$$ $$(4m^2 - 4m + 1) - 4(m^2 - 4) = 0$$

• Step 3: Simplify and solve for $m$:

$$4m^2 - 4m + 1 - 4m^2 + 16 = 0$$

Notice the quadratic terms $4m^2$ cancel out:

$$-4m + 17 = 0 \implies 4m = 17 \implies m = \frac{17}{4}$$

• Step 4: Verify non-zero leading coefficient: $a = m - 2 = \frac{17}{4} - 2 = \frac{9}{4} \neq 0$. The condition is satisfied.
• Conclusion: The required value of $m$ is $\frac{17}{4}$ (or $4.25$).

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Worked Example 3: Operational Optimization via Parabolic Extrema

Problem: In a fictional quadratic model, the hourly expenditure $C(v)$ (in ₹) of hauling a freight rake over a 200 km railway sector varies with sectional speed $v$ (in km/h) according to the quadratic relation:

$$C(v) = 2v^2 - 160v + 4500$$

Find: (a) The optimal operational speed $v$ that minimizes hourly running costs, and (b) The minimum cost per hour.

• Step 1: Identify the quadratic parameters: In $C(v) = av^2 + bv + c$, $a = 2$, $b = -160$, and $c = 4500$. Since $a = 2 > 0$, the parabola opens upward, confirming that the vertex represents an absolute minimum.
• Step 2: Calculate the minimizing speed $v$:

$$v_{\text{optimal}} = -\frac{b}{2a} = -\frac{-160}{2(2)} = \frac{160}{4} = 40 \text{ km/h}$$

• Step 3: Compute the minimum hourly expenditure: Substitute $v = 40$ into the cost function:

$$C(40) = 2(40)^2 - 160(40) + 4500 = 2(1600) - 6400 + 4500$$ $$C(40) = 3200 - 6400 + 4500 = -3200 + 4500 = 1300 \text{ INR/hour}$$

Alternatively, use the vertex ordinate formula:

$$C_{\min} = \frac{4ac - b^2}{4a} = \frac{4(2)(4500) - (-160)^2}{4(2)} = \frac{36000 - 25600}{8} = \frac{10400}{8} = 1300 \text{ INR/hour}$$

• Conclusion: The optimal operating speed is 40 km/h with a minimum cost of ₹1,300 per hour.