Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Algebra → Quadratic Equations ↓


Nature of Roots


Quadratic equations are an integral part of algebra, especially when we are dealing with problems involving curves, trajectories, and parabolic shapes. These equations usually take the form ax^2 + bx + c = 0, where a, b and c are constants, and x represents the variable or unknown we are solving for. One of the key concepts in solving these equations is to understand the 'nature of roots'. The roots of a quadratic equation are the values of x that make the equation true (i.e., the equation equals zero).

Quadratic formula

The quadratic formula is a powerful tool that gives us a way to determine the roots of any quadratic equation. The formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

Here, the expression inside the square root, b^2 - 4ac, is known as the discriminant. It is represented by the symbol Δ (delta) and plays an important role in determining the nature of the roots.

Discriminant and nature of roots

The value of the discriminant tells you how many solutions (roots) the quadratic equation has and what is the nature of these roots. Let us explore the different cases:

1. When Δ > 0 (positive differential)

If the discriminant is greater than zero, it means that the quadratic equation has two distinct real roots. In other words, the parabola intersects the x-axis at two distinct points.

Example: x^2 - 5x + 6 = 0

Calculate the discriminant:

Δ = b^2 - 4ac = (-5)^2 - 4 * 1 * 6 = 25 - 24 = 1 > 0

Thus, the quadratic equation x^2 - 5x + 6 = 0 has two distinct real roots.

2. When Δ = 0 (zero differential)

If the discriminant is equal to zero, then the quadratic equation has exactly one real root. This means that the parabola touches the x-axis at exactly one point, which is called the double root or repeated root.

Example: x^2 - 4x + 4 = 0

Calculate the discriminant:

Δ = b^2 - 4ac = (-4)^2 - 4 * 1 * 4 = 16 - 16 = 0

The quadratic equation x^2 - 4x + 4 = 0 has a repeated root.

3. When Δ < 0 (negative differential)

If the discriminant is less than zero, the quadratic equation has no real roots. Instead, it has two complex roots that are conjugates of each other. This implies that the parabola does not intersect the x-axis.

Example: x^2 + 2x + 5 = 0

Calculate the discriminant:

Δ = b^2 - 4ac = (2)^2 - 4 * 1 * 5 = 4 - 20 = -16

The quadratic equation x^2 + 2x + 5 = 0 has two complex roots.

Visual representation

Case 1: Positive discriminant (Δ > 0)

The quadratic equation y = x^2 - 5x + 6 will look like this:

The graph clearly shows two points where the curve intersects the x-axis, indicating two different real roots.

Case 2: Zero discriminant (Δ = 0)

The quadratic equation y = x^2 - 4x + 4 will look like this:

This graph shows the curve touching the x-axis at only one point, which is the double origin.

Case 3: Negative discriminant (Δ < 0)

The quadratic equation y = x^2 + 2x + 5 will look like this:

Here, the parabola never touches the x-axis, which shows the absence of real roots, and implies that the roots are complex.

Practical example

Example A

Find the nature of the roots of the quadratic equation 2x^2 - 3x + 1 = 0.

Calculate the discriminant:

Δ = b^2 - 4ac = (-3)^2 - 4 * 2 * 1 = 9 - 8 = 1 > 0

The discriminant is greater than zero, so the quadratic equation has two different real roots.

Example B

Determine the nature of the roots for x^2 + 4x + 4 = 0.

Δ = b^2 - 4ac = (4)^2 - 4 * 1 * 4 = 16 - 16 = 0

The discriminant is zero, indicating a double root or repeated root.

Example C

Find the nature of the roots of x^2 + 2x + 10 = 0.

Δ = b^2 - 4ac = (2)^2 - 4 * 1 * 10 = 4 - 40 = -36

The discriminant is less than zero, which shows that the equation has two complex roots (conjugate).

Conclusion

Understanding the nature of roots is an important aspect in solving quadratic equations. It not only tells us the number of solutions of the quadratic equation but also whether these solutions are real numbers or complex numbers. Using the discriminant, we can analyze and predict the nature of these roots even before solving the equation completely.

Understanding these concepts aids in a deeper understanding of quadratic functions and their graphs, which provides a solid foundation for exploring more complex algebraic topics in the future. This understanding of the nature of roots goes beyond solving algebraic equations, as it applies to a range of mathematical problems, helping students appreciate the beauty and utility of mathematics.


Grade 10 → 2.3.3


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Nature of Roots

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