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 ↓


Completing the Square Method


The method of completing the square is a technique used to solve quadratic equations. It transforms the quadratic equation into a form that can be easily solved by taking the square root of both sides. This method is especially useful when you cannot easily factor the quadratic equation or when the equation is complex. The essence of completing the square is to create a perfect square trinomial from the quadratic equation and then solve for the unknown variable.

Understanding quadratic equations

A quadratic equation is an equation of the form:

    ax^2 + bx + c = 0

where a, b, and c are constants, and a ≠ 0 the graph of a quadratic equation is a parabola.

Method of completing the square

To complete the square, our goal is to express the quadratic part of the equation, ax^2 + bx, as a perfect square trinomial, which is an algebraic expression that can be written in the form (x + p)^2.

Step-by-step guide

Step 1: Move the constant term to the right

Begin by rewriting the equation, moving the constant term c to the other side of the equation:

    ax^2 + bx = -c

Step 2: Divide each term by 'a'

If a ≠ 1, divide the entire equation by a to simplify the quadratic coefficient to 1:

    x^2 + (b/a)x = -c/a

Step 3: Complete the square

To complete the square, take half of the coefficient of x, which is (b/a), and square it. Add and subtract this square inside the equation:

    x^2 + (b/a)x + (b/2a)^2 = (b/2a)^2 - c/a

The expression on the left-hand side can now be written as a perfect square.

Step 4: Convert to a perfect square trinomial

Rewrite the left side of the equation as the square of a binomial:

    (x + b/2a)^2 = (b/2a)^2 - c/a

Step 5: Solve for 'x'

Take the square root of both sides of the equation:

    x + b/2a = ±√((b/2a)^2 - c/a)

Finally, solve for x:

    x = -b/2a ± √((b/2a)^2 - c/a)

Examples of completing the square

Example 1

Solve the quadratic equation x^2 + 6x + 8 = 0 by completing the square.

Solution:

  1. Move the constant term to the other side: 
                x^2 + 6x = -8
  2. Divide each term by the coefficient of x^2 (in this case it's 1, so we can skip this step).
  3. Complete the square:
    • Take half of 6 which is 3, and then square it to get 9.
    • Add and subtract 9 inside the equation: 
                          x^2 + 6x + 9 = -8 + 9
  4. Express as a perfect square trinomial: 
                (x + 3)^2 = 1
  5. Solve for x by taking the square root: 
                x + 3 = ±1
  6. Subtract 3 from both sides to find the value of x: 
                x = -3 ± 1
    Thus, x = -2 or x = -4.

Example 2

Solve the quadratic equation 2x^2 + 8x + 5 = 0 by completing the square.

Solution:

  1. Move the constant term to the other side: 
                2x^2 + 8x = -5
  2. Divide each term by 2: 
                x^2 + 4x = -5/2
  3. Complete the square:
    • Take half of 4 which is 2, and then square it to get 4.
    • Add and subtract 4 inside the equation: 
                          x^2 + 4x + 4 = 4 - 5/2
  4. Express as a perfect square trinomial: 
                (x + 2)^2 = 8/2 - 5/2
  5. Simplify to the right: 
                (x + 2)^2 = 3/2
  6. Solve for x by taking the square root: 
                x + 2 = ±√(3/2)
  7. Subtract 2 from both sides: 
                x = -2 ± √(3/2)
    Therefore, x = -2 + √(3/2) or x = -2 - √(3/2).

Visual example

Below is the representation of completing the square for the equation x^2 + 4x + 4 = 0:

x^2 4 x 4 = (x + 2)^2

In the visual above, the large square represents x^2, the rectangle represents 4x term, and the small square represents the number 4, which completes the square. The perfect square trinomial forms a perfect square visual, which represents transformation.

Benefits of completing the square

Completing the square is a versatile method that is beneficial in many mathematical scenarios. Some of its advantages are:

  • It can solve any quadratic equation, even those that cannot be easily factored.
  • It converts quadratic expressions into a format that is easier to analyze and use to graph.
  • This technique helps in deriving the quadratic formula.

Conclusion

Understanding the method of completing the square involves recognizing the ability to reformulate a quadratic equation into a perfect square form. This understanding paves the way for deeper insights into algebra and prepares students for higher-level mathematics. Although the method may seem complicated initially, practice with examples and visual aids makes it an accessible and powerful tool in solving quadratic equations.


Grade 10 → 2.3.2.2


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Completing the Square Method

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