Solving For A Number Summed With Its Square To 90 A Mathematical Exploration
In the fascinating realm of mathematics, we often encounter intriguing puzzles that challenge our problem-solving skills. One such puzzle involves finding a number that, when added to its square, results in 90. This may seem like a simple arithmetic problem at first glance, but it requires a deeper understanding of algebraic equations and their solutions.
Delving into the Equation: Setting the Stage for Discovery
To embark on our quest to find this elusive number, we must first translate the problem into a mathematical equation. Let's represent the unknown number with the variable 'x'. The problem states that the sum of this number and its square is 90. We can express this mathematically as:
x + x² = 90
This equation is a quadratic equation, characterized by the presence of a squared term (x²). Quadratic equations often have two solutions, meaning there might be two numbers that satisfy this condition. To solve this equation and uncover the hidden solutions, we need to rearrange it into a standard form and then employ algebraic techniques.
Rearranging the Equation: Preparing for Solution
To bring the equation into a more manageable form, we subtract 90 from both sides, resulting in:
x² + x - 90 = 0
This is the standard form of a quadratic equation, where the terms are arranged in descending order of their exponents. Now, we have several options for solving this equation, including factoring, using the quadratic formula, or completing the square. In this case, factoring appears to be the most straightforward approach.
Factoring the Equation: Unlocking the Solutions
Factoring involves breaking down the quadratic expression into two binomials (expressions with two terms) that, when multiplied together, yield the original expression. To factor x² + x - 90, we need to find two numbers that add up to 1 (the coefficient of the x term) and multiply to -90 (the constant term). After some deliberation, we find that 10 and -9 satisfy these conditions.
Therefore, we can factor the equation as follows:
(x + 10)(x - 9) = 0
This factorization reveals the two possible solutions for x. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Thus, either (x + 10) = 0 or (x - 9) = 0.
Solving for x: Unveiling the Numbers
Solving for x in each of these equations, we get:
x + 10 = 0 => x = -10
x - 9 = 0 => x = 9
Therefore, the two numbers that satisfy the original condition are -10 and 9. Let's verify these solutions:
For x = -10: -10 + (-10)² = -10 + 100 = 90
For x = 9: 9 + 9² = 9 + 81 = 90
Both solutions hold true, confirming our algebraic journey's success.
Embracing the Power of Quadratic Equations
This exploration demonstrates the power of quadratic equations in solving mathematical puzzles. By translating a word problem into an algebraic equation and employing techniques like factoring, we can unlock hidden solutions and gain a deeper understanding of mathematical relationships. The world of mathematics is filled with such intriguing puzzles, waiting to be unraveled through the application of algebraic principles.
Mathematical problems often present us with intriguing puzzles, challenging us to find numbers that satisfy specific conditions. One such problem involves determining a number that, when added to its own square, yields a sum of 90. This type of problem requires us to delve into the realm of algebra and utilize our equation-solving skills.
Setting Up the Equation: The Foundation of Our Solution
To embark on our mathematical journey, we must first translate the problem's verbal description into a mathematical equation. Let's represent the unknown number with the variable 'x'. The problem states that the sum of this number and its square equals 90. This can be expressed mathematically as:
x + x² = 90
This equation is a quadratic equation, characterized by the presence of a term with the variable raised to the power of two (x²). Quadratic equations are known for their potential to have two distinct solutions, meaning there might be two numbers that satisfy the given condition. To find these solutions, we need to manipulate the equation into a standard form and then apply appropriate algebraic techniques.
Transforming the Equation: Preparing for the Next Step
To make the equation more manageable, we rearrange it by subtracting 90 from both sides. This results in:
x² + x - 90 = 0
This is the standard form of a quadratic equation, where the terms are arranged in descending order of their exponents. Now, we have several options for solving this equation, including factoring, employing the quadratic formula, or completing the square. In this particular case, factoring appears to be the most efficient approach.
Factoring: Deconstructing the Equation to Find Solutions
Factoring involves breaking down the quadratic expression into two binomials (expressions with two terms) that, when multiplied together, produce the original expression. To factor x² + x - 90, we need to identify two numbers that add up to 1 (the coefficient of the x term) and multiply to -90 (the constant term). After careful consideration, we find that 10 and -9 satisfy these conditions.
Therefore, we can factor the equation as follows:
(x + 10)(x - 9) = 0
This factorization is a crucial step, as it reveals the potential solutions for x. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Thus, either (x + 10) = 0 or (x - 9) = 0.
Solving for x: Unveiling the Hidden Numbers
Now, we can solve for x in each of these equations:
x + 10 = 0 => x = -10
x - 9 = 0 => x = 9
Therefore, the two numbers that satisfy the original problem are -10 and 9. To ensure our solutions are correct, let's verify them:
For x = -10: -10 + (-10)² = -10 + 100 = 90
For x = 9: 9 + 9² = 9 + 81 = 90
Both solutions hold true, confirming the accuracy of our algebraic manipulation.
The Power of Quadratic Equations: A Mathematical Tool
This exploration highlights the power of quadratic equations in solving mathematical problems. By translating a verbal problem into an algebraic equation and employing techniques like factoring, we can uncover hidden solutions and gain a deeper understanding of mathematical relationships. The world of mathematics is rich with such intriguing puzzles, waiting to be solved through the application of algebraic principles.
In the realm of mathematics, we encounter problems that challenge our analytical and problem-solving abilities. One such problem involves finding a number that, when added to its square, equals 90. This type of problem necessitates the application of algebraic concepts and techniques.
Translating Words into Math: Setting Up the Equation
The initial step in solving this problem is to translate the given verbal information into a mathematical equation. Let's use the variable 'x' to represent the unknown number. The problem states that the sum of this number and its square is 90. This can be expressed mathematically as:
x + x² = 90
This equation is a quadratic equation, characterized by the presence of a term with the variable raised to the power of two (x²). Quadratic equations are known for their potential to have two distinct solutions. To find these solutions, we need to rearrange the equation into a standard form and then employ appropriate algebraic methods.
Rearranging and Simplifying: Getting Ready to Solve
To bring the equation into a standard form, we subtract 90 from both sides. This results in:
x² + x - 90 = 0
This is the standard form of a quadratic equation, where the terms are arranged in descending order of their exponents. Now, we have several options for solving this equation, including factoring, using the quadratic formula, or completing the square. In this particular case, factoring appears to be the most efficient method.
Factoring: Breaking Down the Equation
Factoring involves expressing the quadratic expression as a product of two binomials (expressions with two terms). To factor x² + x - 90, we need to find two numbers that add up to 1 (the coefficient of the x term) and multiply to -90 (the constant term). After some trial and error, we find that 10 and -9 satisfy these conditions.
Therefore, we can factor the equation as follows:
(x + 10)(x - 9) = 0
This factorization is a critical step in solving the equation. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Thus, either (x + 10) = 0 or (x - 9) = 0.
Solving for the Unknown: Finding the Numbers
Now, we can solve for x in each of these equations:
x + 10 = 0 => x = -10
x - 9 = 0 => x = 9
Therefore, the two numbers that satisfy the original problem are -10 and 9. Let's verify these solutions to ensure accuracy:
For x = -10: -10 + (-10)² = -10 + 100 = 90
For x = 9: 9 + 9² = 9 + 81 = 90
Both solutions hold true, confirming the correctness of our algebraic manipulation.
Quadratic Equations: A Powerful Tool in Mathematics
This exploration showcases the power of quadratic equations in solving mathematical problems. By translating a word problem into an algebraic equation and employing techniques like factoring, we can uncover hidden solutions and gain a deeper appreciation for mathematical relationships. The field of mathematics offers countless such puzzles, waiting to be solved through the application of algebraic principles.