Quadratic Equations Class 10 Mathematics Concept notes and Formula Sheet

 

Concept Notes

1. Definition of a Quadratic Equation

• A quadratic equation is a second-degree polynomial equation of the form: $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants, and $a \neq 0$.

2. Standard Form

• The standard form of a quadratic equation is $ax^2 + bx + c = 0$.

3. Roots of a Quadratic Equation

• The solutions of the quadratic equation are called the roots of the equation. They can be real or complex.
• The roots can be found using various methods, including:
  • Factoring
  • Completing the Square
  • Quadratic Formula

4. Methods for Solving Quadratic Equations

i. Factoring Method

• Factor the quadratic expression into two binomials and solve for $x$: $$ax^2 + bx + c = (px + q)(rx + s) = 0$$ Set each binomial to zero and solve for $x$: $$px + q = 0 \quad \text{and} \quad rx + s = 0$$

ii. Completing the Square Method

• Rewrite the quadratic equation in the form of a perfect square trinomial: $$ax^2 + bx + c = 0 \quad \text{can be rewritten as} \quad \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$
• Solve for $x$ by taking the square root of both sides and isolating $x$.

iii. Quadratic Formula

• The roots of the quadratic equation can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $b^2 - 4ac$ is called the discriminant ($\Delta$).

5. Discriminant

• The discriminant of a quadratic equation is: $$\Delta = b^2 - 4ac$$ It determines the nature of the roots:
  • If $\Delta > 0$, the equation has two distinct real roots.
  • If $\Delta = 0$, the equation has one real root (repeated root).
  • If $\Delta < 0$, the equation has two complex roots (conjugate pairs).

6. Vertex Form of a Quadratic Equation

• The quadratic equation can also be expressed in vertex form: $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex of the parabola.

7. Graph of a Quadratic Equation

• The graph of a quadratic equation is a parabola.
  • If $a > 0$, the parabola opens upwards.
  • If $a < 0$, the parabola opens downwards.
• The axis of symmetry is given by: $$x = -\frac{b}{2a}$$
• The vertex is the minimum or maximum point of the parabola.

8. Relationship Between Roots and Coefficients

• For the quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots ($\alpha$ and $\beta$) are given by:
  • Sum of roots: $\alpha + \beta = -\frac{b}{a}$
  • Product of roots: $\alpha \beta = \frac{c}{a}$

Formula Sheet

1. Quadratic Formula:

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

2. Discriminant:

   $$\Delta = b^2 - 4ac$$

3. Vertex Form:

   $$y = a(x - h)^2 + k$$

   where $(h, k)$ is the vertex.

4. Sum and Product of Roots:

   • Sum of Roots: $$\alpha + \beta = -\frac{b}{a}$$
   • Product of Roots: $$\alpha \beta = \frac{c}{a}$$

5. Vertex of the Parabola:

   $$x = -\frac{b}{2a}$$

   Substitute this value into the quadratic equation to find the y-coordinate of the vertex.

Examples for Practice

1. Solve the quadratic equation $2x^2 - 4x - 6 = 0$ using the quadratic formula.
2. Find the roots of the quadratic equation $x^2 - 5x + 6 = 0$ by factoring.
3. Convert the quadratic equation $x^2 - 4x - 5 = 0$ into its vertex form.
4. Determine the nature of the roots of the quadratic equation $3x^2 + 2x + 1 = 0$ using the discriminant.