Quadratic Equations in One Variable

An equation containing a single variable whose highest power (degree) is $2$ is called a Quadratic Equation in One Variable (একচলবিশিষ্ট দ্বিঘাত সমীকরণ). In standard mathematical notation, it is written as:

Here, $x$ is the unknown variable, $a$ is the coefficient of $x^2$ (leading coefficient), $b$ is the coefficient of $x$, and $c$ is the constant term. The condition $a eq 0$ is essential: if $a = 0$, the $x^2$ term vanishes, reducing the equation to $bx + c = 0$, which is a linear equation of degree $1$.

Quadratic equations are categorized based on whether the linear term ($bx$) is present:

The factorization method relies on the Zero Product Principle: if the product of two real expressions is zero, then at least one of the expressions must be zero:

Step-by-step procedure for splitting the middle term:

1. Express the equation in standard form: $ax^2 + bx + c = 0$.
2. Compute the product of the leading coefficient and the constant term: $P = a \cdot c$.
3. Find two real numbers $p$ and $q$ such that their sum equals the middle coefficient $b$ ($p + q = b$) and their product equals $P$ ($p \cdot q = ac$).
4. Rewrite $bx$ as $px + qx$ and factor by grouping: $(mx + n)(rx + s) = 0$.
5. Set each linear factor equal to zero: $mx + n = 0 \implies x = -n/m$ or $rx + s = 0 \implies x = -s/r$.

When a quadratic expression cannot be easily factored with rational numbers, the method of completing the square converts the equation into the form $(x + k)^2 = d$:

1. Divide through by $a$ ($a eq 0$): $x^2 + rac{b}{a}x + rac{c}{a} = 0$.
2. Transpose the constant term: $x^2 + rac{b}{a}x = -rac{c}{a}$.
3. Add the square of half the coefficient of $x$, namely $\left(rac{b}{2a} ight)^2 = rac{b^2}{4a^2}$, to both sides:
   $$x^2 + 2 \cdot x \cdot rac{b}{2a} + \left(rac{b}{2a} ight)^2 = rac{b^2}{4a^2} - rac{c}{a}$$
4. Write the left side as a perfect square:
   $$\left(x + rac{b}{2a} ight)^2 = rac{b^2 - 4ac}{4a^2}$$
5. Take the square root on both sides:
   $$x + rac{b}{2a} = \pm rac{\sqrt{b^2 - 4ac}}{2a} \implies x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Sridhar Acharya (শ্রীধর আচার্য), the illustrious 9th-century Indian mathematician, provided an ingenious technique to eliminate fractions during completing the square by multiplying the entire equation by $4a$:

The quantity under the radical sign, $b^2 - 4ac$, discriminates between different types of roots and is called the Discriminant, denoted by $D$:

Let $lpha$ and $eta$ be the two roots of $ax^2 + bx + c = 0$ ($a eq 0$). Then:

1. Sum of Roots:

2. Product of Roots:

If $lpha$ and $eta$ are the roots of a quadratic equation, the equation is given by:

Madhyamik exam problems frequently require calculating symmetric values using only $(lpha + eta)$ and $(lpha eta)$:

• $lpha^2 + eta^2 = (lpha + eta)^2 - 2lpha eta = \left(-rac{b}{a} ight)^2 - 2\left(rac{c}{a} ight) = rac{b^2 - 2ac}{a^2}$
• $lpha - eta = \pm \sqrt{(lpha + eta)^2 - 4lpha eta} = \pm rac{\sqrt{b^2 - 4ac}}{a}$
• $rac{1}{lpha} + rac{1}{eta} = rac{lpha + eta}{lpha eta} = rac{-b/a}{c/a} = -rac{b}{c}$
• $lpha^3 + eta^3 = (lpha + eta)^3 - 3lpha eta(lpha + eta) = \left(-rac{b}{a} ight)^3 - 3\left(rac{c}{a} ight)\left(-rac{b}{a} ight) = rac{3abc - b^3}{a^3}$
• $rac{lpha}{eta} + rac{eta}{lpha} = rac{lpha^2 + eta^2}{lpha eta} = rac{(lpha + eta)^2 - 2lpha eta}{lpha eta} = rac{b^2 - 2ac}{ac}$

Translating real-world situations into quadratic equations involves five disciplined steps:

1. Assign Unknown: Let the unknown quantity be $x$, specifying physical units (km/h, minutes, meters, years).
2. Express Other Quantities: Express related quantities in terms of $x$.
3. Formulate Equation: Translate the relationship stated in the problem into an algebraic equation.
4. Solve: Reduce to $ax^2 + bx + c = 0$ and solve by factorization or Sridhar Acharya's formula.
5. Feasibility Check: Reject non-physical roots (negative lengths, negative times, negative speeds, or non-integer digit values).

• Speed-Distance-Time: $ ext{Time} = rac{ ext{Distance}}{ ext{Speed}}$. If speed increases by $v$, time taken decreases: $rac{D}{s} - rac{D}{s + v} = \Delta t$.
• Cistern and Pipe Filling: If tap A fills in $x$ hours, part filled in 1 hour is $rac{1}{x}$. If tap B takes $(x + 5)$ hours, together in 1 hour they fill $rac{1}{x} + rac{1}{x + 5} = rac{1}{T_{ ext{total}}}$.
• Digit Problems: A two-digit number with tens digit $x$ and units digit $y$ is $10x + y$. When digits are reversed, the number becomes $10y + x$.
• Geometric Dimensions: Area of rectangle $= ext{length} imes ext{breadth}$. In right triangles: $ ext{base}^2 + ext{height}^2 = ext{hypotenuse}^2$.