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 ↓


Standard Form of a Quadratic Equation


Quadratic equations are a fundamental part of algebra and are often encountered in both practical and theoretical contexts. It is important to understand them because they appear in various subjects beyond mathematics such as physics, chemistry, and economics. This guide will delve deep into the concept of 'standard form of quadratic equation', providing detailed explanations, textual examples, and visual illustrations for clarity. We will focus on making it as simple as possible for better understanding.

Introduction to quadratic equations

A quadratic equation is a polynomial equation in which a variable is raised to the power of 2. The general form of a quadratic equation is given as:

ax^2 + bx + c = 0

Here, a, b, and c are constants, where a ≠ 0. The variable x represents an unknown. Let's break down each component:

  • a : The coefficient of x^2. This determines the direction of the parabola. If a > 0, the parabola opens upward, and if a < 0, it opens downward.
  • b : The coefficient of x. This affects the position of the parabola along the x-axis.
  • c : The constant term. This moves the parabola up or down along the y-axis, affecting the y-intercept.

Now, let's look at the graphical representation of quadratic functions:

X Y y = ax^2 + bx + c

Standard form in detail

The standard form of a quadratic equation is expressed as:

ax^2 + bx + c = 0

This is the most commonly used form because it is simple and easy to convert into other forms, such as factored form or vertex form.

  • ax^2 term is called a quadratic term.
  • bx term is called a linear term.
  • c term is the constant, or free term, because it does not contain the variable x.

Key characteristics

  • Quadratic term coefficient (a): This determines the width and direction of the parabola. If a is positive, the parabola opens upward, and if negative, it opens downward.
  • Axis of symmetry: The line that divides the parabola into two mirror images. Its equation is x = -b/(2a).

Solving quadratic equations

The standard form is especially useful in finding solutions of quadratic equations, also called roots or zeros. Here are several ways to solve them:

1. Factoring

Factoring involves expressing a quadratic equation as the product of its factors. Consider the equation:

x^2 - 5x + 6 = 0

It can be thought of as follows:

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

According to the zero product property, either x - 2 = 0 or x - 3 = 0, which gives the solutions x = 2 and x = 3.

2. Completing the square

This technique involves rearranging and adjusting the equation to turn it into a perfect square trinomial. For example:

x^2 + 6x + 5 = 0

Rewriting this gives:

(x + 3)^2 - 4 = 0

Solving (x + 3)^2 = 4 gives x + 3 = ±2, which results in x = -1 or x = -5.

3. Quadratic formula

The quadratic formula is derived from the standard form and can solve any quadratic equation:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Let us understand this with an example:

2x^2 + 3x - 2 = 0

Calculating the discriminant: b^2 - 4ac = 3^2 - 4 * 2 * (-2) = 9 + 16 = 25

Solve for x using the formula:

x = (-3 ± sqrt(25)) / 4 => x = 1/2 or x = -2

Visual representation

Graphs can help to visually understand the roots of a quadratic equation. Below is a basic representation of how the roots are represented as intersections on the x-axis:

x=1/2 x=-2

Applications of quadratic equation

There are many applications of quadratic equations in real life. From calculating the trajectory of an object under gravity to optimizing areas and costs, they play a vital role. Here are some examples:

1. Projectile motion

The motion of an object thrown in the air follows a parabolic path determined by quadratic formulas. Miles wants to calculate where his baseball will fall if it is thrown from a height.

2. Economics

Quadratic equations can show the relationship between cost, revenue, or profit functions in a business scenario. For example, finding the price point to maximize profit may involve solving a quadratic equation.

Conclusion

The standard form of a quadratic equation is important for understanding polynomial equations of degree 2. It makes it easy to solve using various methods and shows the basic properties of a quadratic equation. Through the visualizations and various solution techniques presented, mastery of quadratic equations becomes accessible for both practical and theoretical applications. With time and practice, understanding the implications of the standard form becomes an important mathematical skill.


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Standard Form of a Quadratic Equation

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