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 ↓


Factorization Method


The factorization method is a technique used to solve quadratic equations. A quadratic equation is an equation of this form:

ax^2 + bx + c = 0

where a, b, and c are constants, and x is the variable to be solved for. The goal of factoring is to express the quadratic equation as the product of two binomials.

Understanding quadratic equations

To fully understand the factorization method, let's first look at what a quadratic equation looks like. Consider the equation:

x^2 + 5x + 6 = 0

This is a simple quadratic equation. Here, a = 1, b = 5, and c = 6.

Steps to solve by factorization

Solving a quadratic equation by factorization involves these main steps:

Step 1: Identify the equation

First, make sure the quadratic equation is in the standard form ax^2 + bx + c = 0.

Step 2: Factor the quadratic

To factor out the quadratic factor, we need two numbers, say m and n, such that:

  • m * n = a * c
  • m + n = b

Step 3: Write as a product of binomials

Express the quadratic equation as follows:

(x + m)(x + n) = 0

Step 4: Solve for x

Set each factor equal to zero and solve for x:

x + m = 0
x + n = 0

Example 1: Solving x^2 + 5x + 6 = 0

Let us apply the factorization method to solve:

x^2 + 5x + 6 = 0

  1. Identify a = 1, b = 5, c = 6.

  2. We need two numbers that multiply by a * c = 1 * 6 = 6 and add to b = 5.

    • Possible pairs: (2, 3) because 2 * 3 = 6 and 2 + 3 = 5.

  3. Rewrite the equation using these factors:

    x^2 + 2x + 3x + 6 = 0

  4. Group the terms and factor:

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

  5. Find the common factor (x + 2):

    (x + 2)(x + 3) = 0

  6. Set each factor to zero and solve for x:

    x + 2 = 0 => x = -2
    x + 3 = 0 => x = -3

The solutions are x = -2 and x = -3.

Example 2: Solving 2x^2 + 7x + 3 = 0

Now, let us consider another example:

2x^2 + 7x + 3 = 0

  1. Here, a = 2, b = 7, c = 3.

  2. We need two numbers whose product is a * c = 2 * 3 = 6 and the sum is b = 7.

    • Possible pairs: (6, 1) because 6 * 1 = 6 and 6 + 1 = 7.

  3. Rewrite the equation using these numbers:

    2x^2 + 6x + 1x + 3 = 0

  4. Group terms and factors:

    (2x^2 + 6x) + (1x + 3) = 0
    2x(x + 3) + 1(x + 3) = 0

  5. Find the common factor (x + 3):

    (2x + 1)(x + 3) = 0

  6. Set each factor to zero and solve for x:

    2x + 1 = 0 => 2x = -1 => x = -1/2
    x + 3 = 0 => x = -3

The solutions are x = -1/2 and x = -3.

Example 3: Solving 3x^2 - 12x + 12 = 0

Finally, let's take the quadratic equation:

3x^2 - 12x + 12 = 0

  1. Here, a = 3, b = -12, c = 12.

  2. Required product: a * c = 36 and required sum: -12.

    • Possible pairs: (-6, -6) because -6 * -6 = 36 and -6 + -6 = -12.

  3. Rewrite the equation:

    3x^2 - 6x - 6x + 12 = 0

  4. Group terms and factors:

    (3x^2 - 6x) - (6x - 12) = 0
    3x(x - 2) - 6(x - 2) = 0

  5. Find the common factor (x - 2):

    (3x - 6)(x - 2) = 0

  6. Set each factor to zero and solve for x:

    3x - 6 = 0 => 3x = 6 => x = 2
    x - 2 = 0 => x = 2

Its solution is x = 2 (a repeated root).

Visual representation

Visualization can help in understanding factorization. Below is a representation of the factorization of x^2 + 5x + 6.

x^2 + 5x + 6 , 0

Break it down into the following factors:

(x + 2) (x + 3)

You can see how the expression x^2 + 5x + 6 (x + 2)(x + 3) is broken down into factors.

Special cases

Sometimes, factoring involves special patterns such as difference of squares, perfect square trinomials, or is not possible with real numbers.

Difference of squares

A quadratic in the form x^2 - y^2 can be factored as (x + y)(x - y).

Example: x^2 - 49 = (x + 7)(x - 7)

Perfect square trinomial

A quadratic in the form (x + a)^2 = x^2 + 2ax + a^2 is a perfect square trinomial.

Example: x^2 + 6x + 9 = (x + 3)^2

Non-factorizable quadrilaterals

Sometimes, the integers m and n may not exist to satisfy the multiplication-sum conditions, especially if complex numbers are required. In such cases, alternative methods such as completing the square or using the quadratic formula can be used. They are helpful.

The factorization method is effective for solving many quadratic equations, especially when it is possible to decompose into integer factors.

Practice

  1. Factor out x^2 - 9x + 18 = 0 and solve.
  2. Solve by factorization: 2x^2 + x - 1 = 0.
  3. Factor out 3x^2 - x - 2 = 0 and find x.
  4. Determine the solutions for x^2 + 4x + 4 = 0 by using factorization.
  5. Using factorization, solve x^2 - 4 = 0.

Practising these examples will help to strengthen the understanding of the factorisation method in solving quadratic equations.

Factorization method provides an elegant and simple approach to solve quadratic equations by converting them into product of linear factors and then finding their roots.


Grade 10 → 2.3.2.1


U
username
0%
completed in Grade 10

← Prev (2.3.2)
Methods of Solving Quadratic Equations
Next (2.3.2.2) →
Completing the Square Method

Comments 



Factorization Method

https://www.buddymath.com/g10/2.3.2.1?bmal=en