Find The Area Of A Rhombus With Diagonals Of Lengths 10 Cm And 8.2 Cm.2. Find The Value Of ( 3 0 + 4 − 1 ) × 2 2 \left(3^0 + 4^{-1}\right) \times 2^2 ( 3 0 + 4 − 1 ) × 2 2 .3. Factorize: 7 A 2 + 14 A 7a^2 + 14a 7 A 2 + 14 A .4. Find The Value Of M M M For Which $2^m \div 2^{-3} =

Introduction

Mathematics is a fundamental subject that plays a crucial role in our daily lives. It is a subject that deals with numbers, quantities, and shapes, and it is used to solve problems in various fields such as science, engineering, economics, and finance. In this article, we will discuss four mathematical problems that will help you improve your problem-solving skills.

Problem 1: Finding the Area of a Rhombus

A rhombus is a quadrilateral with all sides of equal length. The diagonals of a rhombus bisect each other at right angles. To find the area of a rhombus, we need to know the lengths of its diagonals.

Step 1: Understand the Problem

The problem asks us to find the area of a rhombus with diagonals of lengths 10 cm and 8.2 cm.

Step 2: Recall the Formula

The formula to find the area of a rhombus is:

Area = (d1 × d2) / 2

where d1 and d2 are the lengths of the diagonals.

Step 3: Plug in the Values

Substitute the values of the diagonals into the formula:

Area = (10 × 8.2) / 2

Step 4: Calculate the Area

Calculate the area by multiplying the diagonals and dividing by 2:

Area = 82 / 2 Area = 41

Therefore, the area of the rhombus is 41 square centimeters.

Problem 2: Evaluating an Expression

The problem asks us to evaluate the expression $\left(3^0 + 4^{-1}\right) \times 2^2$.

Step 1: Understand the Problem

The problem asks us to evaluate an expression that involves exponentiation and multiplication.

Step 2: Recall the Order of Operations

The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Step 3: Evaluate the Exponents

Evaluate the exponents in the expression:

$3^0 = 1$ $4^{-1} = \frac{1}{4}$

Step 4: Add the Results

Add the results of the exponents:

$1 + \frac{1}{4} = \frac{5}{4}$

Step 5: Multiply by 2^2

Multiply the result by $2^2$:

$\frac{5}{4} \times 4 = 5$

Therefore, the value of the expression is 5.

Problem 3: Factorizing an Expression

The problem asks us to factorize the expression $7a^2 + 14a$.

Step 1: Understand the Problem

The problem asks us to factorize an expression that involves a quadratic term and a linear term.

Step 2: Recall the Factorization Formula

The formula to factorize a quadratic expression is:

$a^2 + 2ab + b^2 = (a + b)^2$

Step 3: Factorize the Expression

Factorize the expression by identifying the greatest common factor (GCF) of the terms:

$7a^2 + 14a = 7a(a + 2)$

Therefore, the factorized form of the expression is $7a(a + 2)$.

Problem 4: Solving an Equation

The problem asks us to find the value of $m$ for which $2^m \div 2^{-3} = 1$.

Step 1: Understand the Problem

The problem asks us to solve an equation that involves exponentiation and division.

Step 2: Recall the Quotient Rule

The quotient rule states that when we divide two numbers with the same base, we subtract the exponents:

$a^m \div a^n = a^{m-n}$

Step 3: Apply the Quotient Rule

Apply the quotient rule to the equation:

$2^m \div 2^{-3} = 2^{m-(-3)}$ $2^m \div 2^{-3} = 2^{m+3}$

Step 4: Set Up the Equation

Set up the equation by equating the exponents:

$2^{m+3} = 1$

Step 5: Solve for m

Solve for $m$ by setting the exponent equal to 0:

$m + 3 = 0$ $m = -3$

Therefore, the value of $m$ is -3.

Conclusion

In this article, we have discussed four mathematical problems that will help you improve your problem-solving skills. We have learned how to find the area of a rhombus, evaluate an expression, factorize an expression, and solve an equation. By practicing these problems, you will become more confident and proficient in solving mathematical problems.

Q&A: Frequently Asked Questions

In this section, we will answer some frequently asked questions related to mathematical problem-solving.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable. For example, $x^2 + 4x + 4 = 0$ is a quadratic equation. A linear equation, on the other hand, is a polynomial equation of degree one, which means it has a variable but no squared variable. For example, $2x + 3 = 0$ is a linear equation.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, to factorize the expression $x^2 + 6x + 8$, you need to find two numbers whose product is 8 and whose sum is 6. The numbers are 4 and 2, so the expression can be factored as $(x + 4)(x + 2)$.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., it can be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b$ is not zero. For example, $\frac{1}{2}$ and $\frac{3}{4}$ are rational numbers. An irrational number, on the other hand, is a number that cannot be expressed as the ratio of two integers. For example, $\sqrt{2}$ and $\pi$ are irrational numbers.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to find the values of the variables that satisfy all the equations in the system. There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. For example, to solve the system of equations $x + y = 2$ and $x - y = 1$, you can use the substitution method by solving one equation for one variable and substituting the result into the other equation.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A function must assign to each input exactly one output. For example, the function $f(x) = 2x$ assigns to each input $x$ exactly one output $2x$. A relation, on the other hand, is a set of ordered pairs that do not necessarily assign to each input exactly one output. For example, the relation $\{(1, 2), (2, 3), (3, 1)\}$ does not assign to each input exactly one output.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on the coordinate plane that satisfy the function. You can use a graphing calculator or a computer program to graph a function. Alternatively, you can use the following steps to graph a function:

1. Determine the domain and range of the function.
2. Find the x-intercepts of the function by setting the function equal to zero and solving for x.
3. Find the y-intercept of the function by setting x equal to zero and solving for y.
4. Plot the points on the coordinate plane that satisfy the function.
5. Draw a smooth curve through the points to obtain the graph of the function.

Q: What is the difference between a linear function and a quadratic function?

A: A linear function is a function of the form $f(x) = mx + b$, where $m$ and $b$ are constants. A quadratic function, on the other hand, is a function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. Linear functions have a straight line graph, while quadratic functions have a parabolic graph.

Q: How do I find the equation of a circle?

A: To find the equation of a circle, you need to know the center and radius of the circle. The equation of a circle with center $(h, k)$ and radius $r$ is given by $(x - h)^2 + (y - k)^2 = r^2$. For example, the equation of a circle with center $(2, 3)$ and radius $4$ is $(x - 2)^2 + (y - 3)^2 = 16$.

Q: What is the difference between a sine function and a cosine function?

A: A sine function is a function of the form $f(x) = \sin x$, where $x$ is the angle in radians. A cosine function, on the other hand, is a function of the form $f(x) = \cos x$, where $x$ is the angle in radians. Sine and cosine functions are periodic functions that have a period of $2\pi$. They are used to model periodic phenomena such as sound waves and light waves.

Q: How do I find the equation of a parabola?

A: To find the equation of a parabola, you need to know the vertex and the direction of the parabola. The equation of a parabola with vertex $(h, k)$ and direction $a$ is given by $y - k = a(x - h)^2$. For example, the equation of a parabola with vertex $(2, 3)$ and direction $2$ is $y - 3 = 2(x - 2)^2$.

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer. A rational function, on the other hand, is a function of the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$. Polynomial functions have a finite number of roots, while rational functions can have an infinite number of roots.

Q: How do I find the equation of a line?

A: To find the equation of a line, you need to know the slope and the y-intercept of the line. The equation of a line with slope $m$ and y-intercept $b$ is given by $y = mx + b$. For example, the equation of a line with slope $2$ and y-intercept $3$ is $y = 2x + 3$.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality of the form $ax + b < c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality of the form $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$, where $a$, $b$, and $c$ are constants. Linear inequalities have a linear boundary, while quadratic inequalities have a parabolic boundary.

Q: How do I solve a system of nonlinear equations?

A: To solve a system of nonlinear equations, you need to find the values of the variables that satisfy all the equations in the system. There are several methods to solve a system of nonlinear equations, including substitution, elimination, and numerical methods. For example, to solve the system of equations $x^2 + y^2 = 4$ and $x + y = 2$, you can use the substitution method by solving one equation for one variable and substituting the result into the other equation.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A function must assign to each input exactly one output. For example, the function $f(x) = 2x$ assigns to each input $x$ exactly one output $2x$. A relation, on the other hand, is a set of ordered pairs that do not necessarily assign to each input exactly one output. For example, the relation $\{(1, 2), (2, 3), (3, 1)\}$ does not assign to each input exactly one output.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on the coordinate plane that satisfy the function. You can use a graphing calculator or a computer program to graph a function. Alternatively, you can use the following steps to graph a function:

1. Determine the domain and range of the function.
2. Find the x-intercepts of the function by setting the function equal to zero and solving for x.
3. Find the y-intercept of the function by setting x equal to zero and solving for y.
4. Plot the points on the coordinate plane that satisfy the function.
5. Draw a smooth curve through the points to obtain the graph