NCERT · Class 10 · Mathematics · Quadratic EquationsExplain the concept of Quadratic Equations in detail. Discuss their standard form, methods of finding their roots (factorization and quadratic formula), and the significance of the discriminant in determining the nature of roots.

Comprehensive Study of Quadratic Equations

1. Introduction and Standard Form

• A quadratic equation in variable $x$ is an algebraic equation of the second degree.
• The standard form of a quadratic equation is given by: $$ax^2 + bx + c = 0$$ where $a, b,$ and $c$ are real numbers, and the critical condition is that $a \neq 0$. If $a = 0$, the equation becomes a linear equation.
• Examples include $2x^2 - 3x + 1 = 0$ and $x^2 - 5 = 0$.

2. Methods of Finding Roots (Solutions)

• Factorization Method (Splitting the Middle Term):

  • In this method, we express the quadratic expression $ax^2 + bx + c$ as a product of two linear factors $(px + q)(rx + s)$.
  • We split the middle term $bx$ into two parts such that their sum is $b$ and their product equals $ac$.
  • Setting each factor equal to zero yields the roots of the equation.

• Quadratic Formula (Sridharacharya Formula):

  • When factorization is difficult or impractical, roots can always be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
  • This formula is derived by the method of completing the squares.

3. Discriminant and Nature of Roots

• The expression inside the square root, $b^2 - 4ac$, is called the discriminant (denoted by $D$ or $\Delta$): $$D = b^2 - 4ac$$
• The discriminant dictates the nature of the roots without actually calculating them:
  • Case 1: If $D > 0$, the equation has two distinct real roots given by $\frac{-b + \sqrt{D}}{2a}$ and $\frac{-b - \sqrt{D}}{2a}$.
  • Case 2: If $D = 0$, the equation has two equal real roots (or one repeated real root) given by $\frac{-b}{2a}$.
  • Case 3: If $D < 0$, the equation has no real roots (the roots are complex/imaginary numbers).

4. Real-life Applications

• Quadratic equations are extensively used in physics (projectile motion), engineering, economics (maximizing profit and minimizing cost), and everyday geometry problems involving area and perimeter.