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 ↓


Methods of Solving Quadratic Equations


Quadratic equations are an essential part of algebra, usually taught in Class 10 Mathematics. They form the basis for more advanced topics in mathematics and are also applicable in various real-world contexts such as physics, engineering and finance. A quadratic equation is generally expressed in the form:

ax^2 + bx + c = 0

where a, b, and c are constants, and x represents the unknown variable we want to solve for. The main methods for solving quadratic equations are as follows:

  1. Factoring
  2. Completing the square
  3. Using the quadratic formula
  4. Graphing

1. Factoring

Factoring involves rewriting the quadratic expression in a form that can be set to zero, making it easier to solve for x. The factored form of a quadratic is generally:

(px + q)(rx + s) = 0

Let us consider a simple example:

Example:

Solve the quadratic equation x^2 - 5x + 6 = 0 by factoring.

To factor out, find two numbers that multiply by 6 (the constant term, c) and add up to -5 (x, the coefficient of b). The numbers -2 and -3 meet these conditions:

x^2 - 5x + 6 = (x - 2)(x - 3) = 0

Setting each factor to zero, we get:

x - 2 = 0 or x - 3 = 0

Solving these gives the solutions x = 2 and x = 3.

2. Completing the square

Completing the square turns a quadratic equation into a perfect square trinomial, making it easier to solve. This process involves creating a square by adding and subtracting a specific value. Let's look at this process step-by-step:

Example:

Solve the quadratic equation x^2 + 6x + 5 = 0 by completing the square.

  1. Move the constant term to the other side: 
    x^2 + 6x = -5
  2. Take half of the coefficient of x, square it, and add it to both sides: 
    (6/2)^2 = 3^2 = 9
    x^2 + 6x + 9 = -5 + 9
    The simplification of which is as follows: 
    (x + 3)^2 = 4
  3. Take the square root of both sides: 
    x + 3 = ±2
  4. Find x: 
    x = -3 ± 2
    5. This gives us x = -1 and x = -5.

3. Quadratic formula

The quadratic formula is probably the most robust method for solving any quadratic equation, even when factoring or completing the square is difficult. The formula is given as:

x = (-b ± √(b^2 - 4ac)) / (2a)

Example:

Use the quadratic formula to solve the equation 2x^2 + 4x - 6 = 0.

  1. Identify the coefficients: 
    a = 2, b = 4, c = -6
  2. Substitute into the formula: 
    x = (-4 ± √(4^2 - 4(2)(-6))) / (2 * 2)
  3. Simplify inside the square root: 
    x = (-4 ± √(16 + 48)) / 4
    x = (-4 ± √64) / 4
  4. Calculate the roots: 
    x = (-4 ± 8) / 4
    5. This leads to the following solutions: 
    x = 1 and x = -3

4. Graphing

By graphing a quadratic equation, its solutions or roots can be clearly demonstrated where the parabola intersects the x-axis. A quadratic equation is usually plotted as y = ax^2 + bx + c. Let's consider the equation y = x^2 - x - 6.

(-2,0) (3,0)

The above graph shows the parabola intersecting the x-axis at x = -2 and x = 3, which are solutions of the equation x^2 - x - 6 = 0.

Conclusion

In conclusion, there are several methods one can use to solve quadratic equations: factoring, completing the square, using the quadratic formula, and graphing. Each technique has its own merits and scenarios where it is most applicable. While factoring is efficient for simple quadratic equations, completing the square and the quadratic formula provide systematic approaches that work for more complex quadratics. Graphing provides visual insight into the solutions and the nature of parabolas. Mastering these methods is important for solving real-world problems and advancing in mathematical studies.


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Standard Form of a Quadratic Equation
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Factorization Method

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Methods of Solving Quadratic Equations

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