Solutions Of Quadratic Equation X² = -7x - 8
In the realm of mathematics, quadratic equations hold a prominent position, frequently encountered in various fields ranging from physics and engineering to economics and computer science. These equations, characterized by their second-degree polynomial form, often present a captivating challenge to solve. In this comprehensive guide, we embark on a journey to unravel the intricacies of solving the quadratic equation x² = -7x - 8, delving into the methods, interpretations, and significance of its solutions.
Understanding the Quadratic Equation
Before we delve into the solution process, let's first establish a solid understanding of the quadratic equation itself. A quadratic equation is a polynomial equation of the second degree, meaning that the highest power of the variable (in this case, 'x') is 2. The general form of a quadratic equation is expressed as:
ax² + bx + c = 0
where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. The solutions to a quadratic equation, also known as its roots, are the values of 'x' that satisfy the equation, making the expression equal to zero.
Our specific equation, x² = -7x - 8, can be rearranged into the standard quadratic form by adding 7x and 8 to both sides, resulting in:
x² + 7x + 8 = 0
Now, we have a quadratic equation in the familiar standard form, where a = 1, b = 7, and c = 8. With this foundation in place, we can explore various methods to determine the solutions for 'x'.
Methods for Solving Quadratic Equations
Several approaches exist for solving quadratic equations, each with its own strengths and suitability depending on the specific equation. We'll focus on two primary methods: factoring and the quadratic formula.
1. Factoring
Factoring involves expressing the quadratic equation as a product of two linear factors. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. Let's attempt to factor our equation, x² + 7x + 8 = 0.
We seek two numbers that multiply to 8 (the constant term) and add up to 7 (the coefficient of the 'x' term). After some consideration, we identify the numbers 1 and 8 as potential candidates, as 1 * 8 = 8 and 1 + 8 = 9. However, these numbers don't quite match our required sum of 7. Let's try 2 and 4, 2 * 4 = 8 and 2 + 4 = 6. Still not quite. It seems that factoring this equation directly is not straightforward. Therefore, let's explore the alternative method – the quadratic formula.
2. The Quadratic Formula
The quadratic formula provides a universal solution for any quadratic equation, regardless of its factorability. This powerful formula is derived from completing the square on the general quadratic equation and is expressed as:
x = (-b ± √(b² - 4ac)) / 2a
where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. The symbol '±' indicates that there are two possible solutions, one with addition and one with subtraction.
Let's apply the quadratic formula to our equation, x² + 7x + 8 = 0, where a = 1, b = 7, and c = 8. Substituting these values into the formula, we get:
x = (-7 ± √(7² - 4 * 1 * 8)) / (2 * 1)
Simplifying the expression under the square root:
x = (-7 ± √(49 - 32)) / 2
x = (-7 ± √17) / 2
Therefore, the two solutions for 'x' are:
x₁ = (-7 + √17) / 2
x₂ = (-7 - √17) / 2
These solutions correspond to option B in the given choices, after multiplying the numerator and denominator of each fraction by 2.
Verifying the Solutions
To ensure the accuracy of our solutions, let's substitute each value of 'x' back into the original equation, x² + 7x + 8 = 0, and verify if the equation holds true.
Verification for x₁ = (-7 + √17) / 2
Substituting x₁ into the equation:
((-7 + √17) / 2)² + 7((-7 + √17) / 2) + 8 = 0
Expanding and simplifying:
(49 - 14√17 + 17) / 4 + (-49 + 7√17) / 2 + 8 = 0
(66 - 14√17) / 4 + (-98 + 14√17) / 4 + 32 / 4 = 0
(66 - 14√17 - 98 + 14√17 + 32) / 4 = 0
0 / 4 = 0
0 = 0
The equation holds true for x₁.
Verification for x₂ = (-7 - √17) / 2
Substituting x₂ into the equation:
((-7 - √17) / 2)² + 7((-7 - √17) / 2) + 8 = 0
Expanding and simplifying:
(49 + 14√17 + 17) / 4 + (-49 - 7√17) / 2 + 8 = 0
(66 + 14√17) / 4 + (-98 - 14√17) / 4 + 32 / 4 = 0
(66 + 14√17 - 98 - 14√17 + 32) / 4 = 0
0 / 4 = 0
0 = 0
The equation also holds true for x₂. Therefore, both solutions obtained using the quadratic formula are correct.
Interpreting the Solutions
The solutions to a quadratic equation represent the points where the parabola defined by the equation intersects the x-axis. In our case, the solutions x₁ = (-7 + √17) / 2 and x₂ = (-7 - √17) / 2 are the x-coordinates of the two points where the parabola y = x² + 7x + 8 intersects the x-axis. Since the discriminant (b² - 4ac = 17) is positive, the quadratic equation has two distinct real roots, as we have found.
Conclusion
In this comprehensive exploration, we have successfully navigated the process of solving the quadratic equation x² = -7x - 8. We transformed the equation into the standard quadratic form, explored the factoring method, and ultimately employed the powerful quadratic formula to determine the solutions x₁ = (-7 + √17) / 2 and x₂ = (-7 - √17) / 2. We rigorously verified these solutions and interpreted their significance in the context of the corresponding parabola. This journey underscores the importance of understanding quadratic equations and the diverse methods available for solving them, solidifying our mathematical foundation for tackling more complex challenges.
In this article, we will solve the quadratic equation x² = -7x - 8 using different methods, ensuring a clear understanding of the solution process. Quadratic equations are a fundamental concept in algebra and have widespread applications in various fields, including physics, engineering, and economics. Mastering the techniques for solving them is crucial for any student of mathematics. Let's embark on this step-by-step journey to find the solutions.
Understanding the Problem
The given equation is x² = -7x - 8. Our first step is to rewrite the equation in the standard quadratic form, which is:
ax² + bx + c = 0
To do this, we add 7x and 8 to both sides of the equation:
x² + 7x + 8 = 0
Now we have a quadratic equation in standard form, where a = 1, b = 7, and c = 8. The solutions to this equation, also known as roots, are the values of x that satisfy the equation. We can find these solutions using several methods, including factoring, completing the square, and the quadratic formula.
Method 1: Using the Quadratic Formula
The quadratic formula is a universal method for finding the roots of any quadratic equation. It is given by:
x = (-b ± √(b² - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In our case, a = 1, b = 7, and c = 8. Plugging these values into the formula, we get:
x = (-7 ± √(7² - 4 * 1 * 8)) / (2 * 1)
First, we simplify the expression under the square root:
7² - 4 * 1 * 8 = 49 - 32 = 17
So, the equation becomes:
x = (-7 ± √17) / 2
This gives us two solutions:
x₁ = (-7 + √17) / 2
x₂ = (-7 - √17) / 2
These are the exact solutions of the quadratic equation. We can also approximate these values using a calculator. The solutions can be written separately as:
x₁ ≈ (-7 + 4.123) / 2 ≈ -1.438
x₂ ≈ (-7 - 4.123) / 2 ≈ -5.562
Method 2: Factoring (Attempt)
Factoring involves expressing the quadratic equation as a product of two binomials. If the quadratic equation can be factored, it can provide a quicker solution than the quadratic formula. We are looking for two numbers that multiply to 8 and add up to 7. The factor pairs of 8 are:
- 1 and 8
- 2 and 4
However, none of these pairs add up to 7. Therefore, the equation x² + 7x + 8 = 0 cannot be easily factored using integers. This is why the quadratic formula is a more reliable method for this particular equation.
Method 3: Completing the Square (Brief Overview)
Completing the square is another method for solving quadratic equations. It involves transforming the equation into the form (x + p)² = q, where p and q are constants. While this method can be used for any quadratic equation, it is more complex than the quadratic formula, especially when the coefficients are not integers or when the equation cannot be easily factored.
For our equation x² + 7x + 8 = 0, completing the square would involve the following steps:
- Move the constant term to the right side: x² + 7x = -8
- Add (b/2)² to both sides. In our case, (7/2)² = 49/4: x² + 7x + 49/4 = -8 + 49/4
- Rewrite the left side as a square: (x + 7/2)² = (-32 + 49) / 4
- Simplify: (x + 7/2)² = 17/4
- Take the square root of both sides: x + 7/2 = ± √(17/4)
- Solve for x: x = -7/2 ± √17 / 2, which gives the same solutions as the quadratic formula.
Although completing the square is a valid method, the quadratic formula is generally preferred for equations that are not easily factored.
Verifying the Solutions
To ensure our solutions are correct, we should substitute them back into the original equation x² + 7x + 8 = 0 and verify that the equation holds true.
Verification for x₁ = (-7 + √17) / 2
((-7 + √17) / 2)² + 7((-7 + √17) / 2) + 8
Expanding and simplifying this expression:
((49 - 14√17 + 17) / 4) + ((-49 + 7√17) / 2) + 8
(66 - 14√17) / 4 + (-98 + 14√17) / 4 + 32 / 4
(66 - 14√17 - 98 + 14√17 + 32) / 4
(0) / 4 = 0
The equation holds true for x₁.
Verification for x₂ = (-7 - √17) / 2
((-7 - √17) / 2)² + 7((-7 - √17) / 2) + 8
Expanding and simplifying this expression:
((49 + 14√17 + 17) / 4) + ((-49 - 7√17) / 2) + 8
(66 + 14√17) / 4 + (-98 - 14√17) / 4 + 32 / 4
(66 + 14√17 - 98 - 14√17 + 32) / 4
(0) / 4 = 0
The equation holds true for x₂ as well. Therefore, both solutions are correct.
Conclusion
We have successfully solved the quadratic equation x² = -7x - 8 using the quadratic formula. The solutions are x₁ = (-7 + √17) / 2 and x₂ = (-7 - √17) / 2. We also attempted to solve the equation by factoring, but it was not easily factorable. Additionally, we briefly discussed completing the square as an alternative method. By verifying our solutions, we have confirmed their accuracy. Understanding and applying these methods ensures proficiency in solving quadratic equations, a critical skill in mathematics and related fields. These solutions correspond to option B in the original problem.
Solving quadratic equations is a fundamental skill in algebra, with applications spanning across various scientific and engineering disciplines. A quadratic equation, characterized by its second-degree polynomial form, presents unique challenges and solution methodologies. In this comprehensive exploration, we will dissect the equation x² = -7x - 8, applying diverse techniques to uncover its solutions and providing a robust understanding of the process.
The Essence of Quadratic Equations
At its core, a quadratic equation is a polynomial equation of degree two. The general form of a quadratic equation is represented as:
ax² + bx + c = 0
where 'a', 'b', and 'c' are constants, with 'a' not equal to zero. The solutions to this equation, also termed roots, are the values of 'x' that render the equation true. These roots represent the points where the parabola, described by the quadratic equation, intersects the x-axis.
The equation we aim to solve, x² = -7x - 8, can be transformed into the standard quadratic form by transposing the terms, resulting in:
x² + 7x + 8 = 0
In this form, we can clearly identify the coefficients: a = 1, b = 7, and c = 8. Equipped with this foundational understanding, we can now delve into the various methodologies for determining the solutions for 'x'.
Methodical Approaches to Solving Quadratic Equations
Several methods exist for tackling quadratic equations, each possessing its own advantages and suitability based on the equation's characteristics. We will focus on two primary techniques: the quadratic formula and factoring.
1. Unveiling Solutions with the Quadratic Formula
The quadratic formula stands as a universal tool for solving quadratic equations, irrespective of their factorability. This formula, derived from the process of completing the square on the general quadratic equation, is expressed as:
x = (-b ± √(b² - 4ac)) / 2a
where 'a', 'b', and 'c' correspond to the coefficients of the quadratic equation in its standard form, ax² + bx + c = 0. The '±' symbol signifies the existence of two potential solutions, one involving addition and the other subtraction.
Let's apply this formula to our equation, x² + 7x + 8 = 0, where a = 1, b = 7, and c = 8. Substituting these values, we obtain:
x = (-7 ± √(7² - 4 * 1 * 8)) / (2 * 1)
Simplifying the expression under the square root:
x = (-7 ± √(49 - 32)) / 2
x = (-7 ± √17) / 2
Hence, the two solutions for 'x' are:
x₁ = (-7 + √17) / 2
x₂ = (-7 - √17) / 2
These solutions align with option B in the provided choices.
2. Attempting the Factoring Approach
Factoring involves expressing the quadratic equation as a product of two linear factors. This method hinges on the principle that if the product of two factors equals zero, then at least one of the factors must be zero. Let's attempt to factor our equation, x² + 7x + 8 = 0.
We seek two numbers that multiply to 8 (the constant term) and sum to 7 (the coefficient of the 'x' term). Upon examination, we identify the number pairs (1, 8) and (2, 4) as potential candidates. However, neither pair perfectly satisfies both conditions: 1 * 8 = 8, but 1 + 8 = 9, and 2 * 4 = 8, but 2 + 4 = 6. Therefore, factoring this equation directly proves challenging. This underscores the importance of having alternative methods, such as the quadratic formula, at our disposal.
Solution Verification: Ensuring Accuracy
To ensure the integrity of our solutions, let's substitute each value of 'x' back into the original equation, x² + 7x + 8 = 0, and verify if the equation remains valid.
Verification for x₁ = (-7 + √17) / 2
Substituting x₁ into the equation:
((-7 + √17) / 2)² + 7((-7 + √17) / 2) + 8 = 0
Expanding and simplifying:
(49 - 14√17 + 17) / 4 + (-49 + 7√17) / 2 + 8 = 0
(66 - 14√17) / 4 + (-98 + 14√17) / 4 + 32 / 4 = 0
(66 - 14√17 - 98 + 14√17 + 32) / 4 = 0
0 / 4 = 0
0 = 0
The equation holds true for x₁.
Verification for x₂ = (-7 - √17) / 2
Substituting x₂ into the equation:
((-7 - √17) / 2)² + 7((-7 - √17) / 2) + 8 = 0
Expanding and simplifying:
(49 + 14√17 + 17) / 4 + (-49 - 7√17) / 2 + 8 = 0
(66 + 14√17) / 4 + (-98 - 14√17) / 4 + 32 / 4 = 0
(66 + 14√17 - 98 - 14√17 + 32) / 4 = 0
0 / 4 = 0
0 = 0
The equation holds true for x₂ as well. This confirms the accuracy of both solutions obtained using the quadratic formula.
Interpreting the Significance of the Solutions
The solutions to a quadratic equation represent the x-intercepts of the parabola defined by the equation. In our scenario, the solutions x₁ = (-7 + √17) / 2 and x₂ = (-7 - √17) / 2 denote the x-coordinates of the two points where the parabola y = x² + 7x + 8 intersects the x-axis. The discriminant (b² - 4ac = 17) being positive indicates that the quadratic equation possesses two distinct real roots, as we have successfully determined.
Concluding Thoughts
In this detailed exploration, we have adeptly solved the quadratic equation x² = -7x - 8. We transformed the equation into its standard form, investigated the factoring method, and ultimately employed the quadratic formula to derive the solutions x₁ = (-7 + √17) / 2 and x₂ = (-7 - √17) / 2. We meticulously verified these solutions and elucidated their significance within the context of the corresponding parabola. This endeavor underscores the critical importance of comprehending quadratic equations and the diverse methodologies available for solving them, thereby strengthening our mathematical prowess for tackling more intricate challenges.
