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 Formula


The quadratic formula provides a method for solving quadratic equations. Quadratic equations are those that can be rearranged into the standard form:

ax^2 + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The key to solving these equations lies in the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

This formula allows you to solve any quadratic equation by substituting the values of a, b, and c from the equation into the formula. The result gives you the values of x that satisfy the equation.

Understanding the components of the quadratic formula

Before using the quadratic formula, it is necessary to understand its components:

  • a, b, c - coefficients from the quadratic equation ax^2 + bx + c = 0
  • √ - square root symbol that indicates calculation of the square root of the expression inside
  • ± - plus/minus symbol which means you must solve the equation twice: once with the plus (+) and once with the minus (-)

Step-by-step solution using quadratic formula

Let's take a deeper look at how to apply the quadratic formula step by step with an example. Let's say you have a quadratic equation:

2x^2 + 3x - 2 = 0
  1. Identify a, b, c: From the equation 2x^2 + 3x - 2 = 0, identify:
    • a = 2
    • b = 3
    • c = -2
  2. Substitute the values into the quadratic formula: 
    x = (-3 ± √(3² - 4 * 2 * -2)) / (2 * 2)
  3. Calculate the discriminant. The discriminant is the expression inside the square root: 
    b² - 4ac = 3² - 4 * 2 * -2 = 9 + 16 = 25
  4. Calculate the square root of the discriminant: 
    √25 = 5
  5. Apply the ± operator to find the solution: 
    x = (-3 + 5) / 4 → x = 1/2
    x = (-3 - 5) / 4 → x = -2

Validating the solutions

Once we have found x = 1/2 and x = -2, it is always a good idea to validate these solutions by substituting them back into the original equation to make sure our solutions are correct.

For x = 1/2: 2(1/2)^2 + 3(1/2) - 2 = 2(1/4) + 3/2 - 2 = 1/2 + 3/2 - 2 = 2 - 2 = 0
For x = -2: 2(-2)^2 + 3(-2) - 2 = 2(4) - 6 - 2 = 8 - 6 - 2 = 0

Both solutions returned the correct value, confirming our calculations.

Visual representation of solutions

Understanding the solutions visually can help deepen your understanding of the quadratic formula. Consider the graph of a quadratic equation:

x = 1/2 x = -2

The above graph shows the parabola of the quadratic equation 2x^2 + 3x - 2 = 0. The red points are the roots or solutions of x on the parabola where it intersects x-axis, which corresponds to x = 1/2 and x = -2.

Special cases

Not all quadratic equations have two distinct real number solutions. Depending on the value of the discriminant b² - 4ac, there are three cases to consider:

Case 1: Two distinct real roots

This case occurs when the discriminant is greater than zero (b² - 4ac > 0). This shows that the parabola intersects the x-axis at two different points. The quadratic equation 2x^2 + 3x - 2 = 0 that we solved earlier is an example of this case.

Case 2: A repeated real root

This happens when the discriminant is equal to zero (b² - 4ac = 0), which indicates that the parabola touches the x-axis at exactly one point, also called the repeated origin. For example, the equation:

x^2 - 6x + 9 = 0

Substituting into the quadratic formula:

x = (6 ± √(6² - 4 * 1 * 9)) / (2 * 1) = (6 ± √(36 - 36)) / 2
x = (6 ± 0) / 2 → x = 3

In this case, there is a unique solution, x = 3.

Case 3: No real roots

If the discriminant is less than zero (b² - 4ac < 0), then the quadratic equation has no real roots, which means that the parabola does not intersect the x-axis at any point. For example, in the equation:

x^2 + 4x + 5 = 0

Substituting into the quadratic formula:

x = (-4 ± √(4² - 4 * 1 * 5)) / (2 * 1) = (-4 ± √(16 - 20)) / 2
x = (-4 ± √(-4)) / 2

The negative inside the square root means that there are no real solutions. However, these roots can be represented using complex numbers.

Importance of quadratic formula

The strength of the quadratic formula lies in its ability to solve any quadratic equation, regardless of complexity. It is a powerful algebraic tool that simplifies finding roots, making it essential in mathematics.

Widely used in physics, engineering, finance, and other sciences, the quadratic formula helps predict phenomena such as projectile motion, calculate optimal points of functions, and understand biological and economic evolution through various models.

The alacrity with which the quadratic formula can transform complex problems into manageable solutions demonstrates its indispensability within mathematical education and beyond.

Conclusion

To master the use of the quadratic formula it is necessary to understand not only the arithmetic operations but also its conceptual underpinnings—the way it geometrically represents the solution of quadratic relations in real-world applications.

Remember, verifying each solution obtained through the quadratic formula ensures accuracy and strengthens your journey into the vast landscape of algebra and mathematics.


Grade 10 → 2.3.2.3


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Nature of Roots

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