Math adventure stories: Mystery of vanishing gemstone

Chapter 1: The coded scroll

Math adventure stories : One morning, disaster strikes in the magical village of Algebraville, where everything runs on the power of math.The Infinity Crystal—the heart of the town’s magic  is no longer there! Without it, the town’s floating bridges start to sway, the lights go down, and the fountains stop flowing.

The mayor, Ms. Equation, called an emergency  meeting .

“Citizens of Algebraville,” she said, “without the Infinity Crystal, our town’s magic will fade.” A trail of riddles left by the thief—the myterious Variable Master—could take us to the Crystal. “Who will solve the puzzles and save the town?”

To solve them, the kids must use their math skills. Only the cleverest can unlock the secrets of algebra.

A group of daring children stepped forward. They called themselves the Number Detectives and promised to return the crystal. Their first clue was a rolled-up scroll found on the pedestal. When unrolled, it presented a puzzle.

“To find where the map is hidden, compute the equation 3x + 5 = 14. “What is x?”

Solving the Riddles

The Number detectives huddled together to discuss the issue. “We need to isolate x,” said Mira, the group’s leader. “First, subtract 5 from both sides of the equation.”

• 3x+5=14
• 3x=14-5
• 3x=9

“Now divide both sides by 3,” added Leo.

• x=9/3
• x=3

As soon as they figure out the puzzle, a secret compartment in the library wall clicks open, and the numbers on the scroll glow brightly. They discover a golden map within, but a portion of it is missing. There is a message on the map’s reverse:

“Good job, Number Detectives! To finish this map, head to the Forest of Fractions and solve the next puzzle.”

The children gather their supplies and proceed on their journey, excited to discover what challenges await.

To be continued …

Stay tuned for chapter 2: The Forest of Fractions, where the number detectives encounter a  wise old turtle and tackle fraction puzzles to move forward in their adventure!

Two dimensional geometry formulas  sometimes known as plane geometry formulas , are the branch of geometry concerned with shapes, figures, and their properties in the two-dimensional plane. The plane has two dimensions: length and width, represented by the x-axis (horizontal) and y-axis (vertical) in a Cartesian coordinate system.

The basic ideas of two-dimensional geometry are points, lines, angles, forms (such as triangles, circles, and polygons), and their attributes. Let’s look at some essential aspects:

1. Cartesian Coordinates and Plane

The Cartesian coordinate system is the basis for two-dimensional geometry. The coordinate plane is a flat surface with points specified by a pair of numbers  (x,y).

•  x represents the abscissa (horizontal distance from the origin).
•  y represents the ordinate (vertical distance from the origin).

The origin (0,0) is where the x-axis and y-axis cross.

2. Points and coordinates

• A point is an location in the plane without size or dimension, expressed by coordinates  (x,y).
• The distance between two points (x_1, y_1​) and  (x_2, y_2) is given by the distance formula:

• The midpoint between two points (𝑥_1, 𝑦_ 1)  and (𝑥_ 2, 𝑦_ 2) is the point that divides the segment connecting them into two equal parts.

3. Lines and slopes

A line in the plane can be represented by the following linear equation:

• The slope (𝑚) of a line running through two points (𝑥_1, 𝑦_1)  and (𝑥_2, 𝑦_2)  can be calculated as:

• The equation of a line in slope-intercept form (when the slope $m$ and y-intercept c are known) is:

4. Angles

Angle between two lines: The  angle θ between two lines with slopes of m_1  and 𝑚_2  is as follows:

• The angle between two parallel lines is 0∘, and they both have the same slope.
• The slopes of perpendicular lines are negative reciprocals of one another, meaning that 𝑚_1 𝑚_2 = −1 , and the angle between them is 90∘ .

5. Distance between a point and a line.

The distance between a point P(x_0​,y_0​) and a line Ax+By+C=0 is determined by the formula:

6. Types of 2D Triangle Geometric Figures

A triangle is a three-sided polygon. Triangles can be classified according to angles as follows:

• Acute triangle: All angles less than 90∘
• Right triangle: One angle is 90∘
• Obtuse triangle: One angle is greater than 90∘

A triangle’s area is:

The area of a triangle with vertices A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), and  is as follows:

Quadrilaterals
A four-sided polygon is called a quadrilateral. Among the varieties are:

Square , Rectangular, Parallelogram, trapezium, rhombus
A rectangle’s area is:

Circles
The set of all points in the plane that are equally spaced from a fixed point (the center) is called a circle. The equation for a circle with radius r and center (h,k) is as follows:

The area and circumference of a circle are given by:

Area:

Circumference:

Polygons
A closed figure with straight sides is called a polygon. Among the examples are:

Pentagon (5 sides)
Hexagon (6 sides)
Octagon (8 sides)

The sum of the interior angles of a polygon with $n$ sides is given by:

Sum of interior angles=(n−2)×180∘

7. Conic Sections

Conic sections are curves that can be obtained by intersecting a plane with a cone. Each has a specific general equation.

• Circle:

• Ellipse:

• Parabola:

• Hyperbola:

8. Coordinated Geometry

In Two dimensional geometry formulas , coordinate geometry (also known as analytic geometry) employs the Cartesian coordinate system to algebraically represent and investigate geometric shapes.

For example:

• The distance formula finds the distance between two places.
• The slope of a line indicates its steepness and direction.
• A circle’s equation can be calculated using the distance between the center and a point on the circle.

Partial fraction formula: Integration by Partial Fractions is a method for integrating rational functions that are fractions of polynomials. The goal is to reduce a complex rational function to simpler fractions that are easier to integrate.

Steps for Partial Fraction Integration:

1.  Check to see if the degree of the numerator is less than that of the denominator:

If the degree of the numerator is greater than or equal to the degree of the denominator, use long division to rewrite the integral so that the degree of the numerator is less than that of the denominator.
2. Factor the denominator, if possible:

Try to break the denominator down into irreducible elements (linear or quadratic).
3. Set up the partial fractions: Depending on the factorization of the denominator, break the rational function into simpler fractions:

• For linear factors (x-a) , use the form

• Use the form for repeated linear components  (x−a)^n

• For quadratic factors that cannot be reduced (x^2+bx+c) use the form

• Use the following expression for repeated irreducible quadratic factors: (x^2 + bx + c)^n

4. Solve for the constants: After setting up the partial fractions , multiply both sides of the equation by the common denominator and equate coefficients of like powers of x to solve for the constants.

5. Integrate the simpler fractions: After determining the constants, integrate each term separately.

Partial fraction formula Partial fraction formula

1. Linear factors in the denominator

If the denominator consists of different linear factors:

Example:

Steps

1. Decompose

2. Solve for A and B.

3. Integrate each term.

Solution: A=1, B=1

2. Repeated Linear Factors

If the denominator includes repeated linear factors:

Example:

Steps

1. Decompose

2. Integrate each term.

Solution:

3. Quadratic Factors

If the denominator includes an irreducible quadratic factor:

Example:

1. For irreducible x^2+4, rewrite:

2. Solve for coefficients if needed (A = 0, B= 1)

3. Integrate using

Solution:

4. Repeated Quadratic Factors

For several irreducible quadratic factors:

Example:

Steps

1. Decompose

2. Solve for coefficients.

3. Integrate each term.

Solution:

5. Mixed Linear and Quadratic Factors

When the denominator contains both linear and quadratic factors:

Example:

Steps

1. Decompose

2. Solve for A, B, and C.

3. Integrate each term separately.

Example :

Step 1 : First, configure the partial fractions.

As (x+1)^2 (x−2), the denominator is already factored as (x+1)^2 (x−2). The integrand will therefore be represented as a sum of partial fractions of the following form:

Step 2: Multiply both sides by (𝑥+1)^2 ( 𝑥 − 2 ) 

Step 3: Expand the right side.

Next, expand each phrase on the right-hand side:

Thus we get

Simplifying

Step 4: Compare the coefficients.

We equate the coefficients of powers of 𝑥 on both sides of the equation.

• Coefficient of x^2: A+C=2
• Coefficient of $x$: A+B+2C=0
• Constant term: −2A−2B+C=−1

Step 5: solve the system of equations.

We will now solve the system of equations:

1. A+C=2
2. -A +B+2C=0
3. -2A-2B+2C=-1

From equation 1

C=2-A

Substitute C=2−A into equations (2) and (3):

From equation 2

From equation 3

Now solve equations (4) and (5).

Based on equation (4):

𝐵 = 3 𝐴 − 4

Substitute B=3A−4 into equation (5):

Now substitute A=11/9 inti C = 2-A

Finally , substitute A = 11/9 into B = 3A-4 :

Step 6: Write a partial fraction decomposition.

We currently have:

Step 7: Integrate the terms.

We may now integrate each phrase separately.

Thus, the solution is:

This is the final answer.

Practice problems

Solve the integral using partial fraction

1.

2.

3.

4.

5.