What Are The Solutions To The Equation X² + 5x - 8 = 4x + 4? Options: A. -4 B. -3 C. 4 D. 5 E. 3 F. -2
In the realm of mathematics, quadratic equations stand out as fundamental concepts with wide-ranging applications. These equations, characterized by the presence of a variable raised to the power of two, often present themselves in the form ax² + bx + c = 0, where a, b, and c are constants. Solving quadratic equations involves finding the values of the variable (typically denoted as 'x') that satisfy the equation, also known as the roots or solutions. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graph.
Various methods exist for solving quadratic equations, each with its own strengths and applicability. Factoring, a technique that involves expressing the quadratic expression as a product of two linear factors, is often the first approach considered when dealing with simpler equations. The quadratic formula, a more versatile method, provides a direct solution for any quadratic equation, regardless of its complexity. Completing the square, another technique, involves manipulating the equation to create a perfect square trinomial, which can then be easily solved.
The significance of quadratic equations extends far beyond the realm of pure mathematics. These equations find practical applications in diverse fields, including physics, engineering, economics, and computer science. For instance, in physics, quadratic equations can model projectile motion, describing the trajectory of objects thrown into the air. In engineering, they play a crucial role in designing structures and optimizing processes. In economics, quadratic equations can model supply and demand curves, helping to understand market dynamics. In computer science, they are used in various algorithms and data structures.
In this article, we will delve into the process of solving a specific quadratic equation, x² + 5x - 8 = 4x + 4, by employing algebraic techniques. Our goal is to identify the values of x that satisfy this equation, thereby gaining a deeper understanding of the equation's behavior and its potential applications. We will meticulously walk through each step, providing clear explanations and insights to ensure a comprehensive understanding of the solution process.
Our task at hand is to determine which of the provided options (A. -4, B. -3, C. 4, D. 5, E. 3, F. -2) are solutions to the quadratic equation x² + 5x - 8 = 4x + 4. This involves verifying whether substituting each value for 'x' in the equation results in a true statement. To achieve this, we will systematically substitute each option into the equation and evaluate the result. If the left-hand side (LHS) of the equation equals the right-hand side (RHS) after the substitution, then the value is a solution. This methodical approach allows us to accurately identify the roots of the equation from the given choices.
Before we begin the substitution process, it's crucial to rearrange the equation into its standard quadratic form. This form, ax² + bx + c = 0, facilitates the application of various solving techniques, such as factoring, the quadratic formula, or completing the square. By transforming the equation into this standard form, we create a clear and organized structure that simplifies the subsequent steps in finding the solutions. This preparatory step is essential for ensuring accuracy and efficiency in our problem-solving process.
To transform the equation x² + 5x - 8 = 4x + 4 into standard form, we need to consolidate all terms on one side of the equation, leaving zero on the other side. This involves subtracting 4x and 4 from both sides of the equation. By performing these operations, we maintain the equality of the equation while simultaneously rearranging its terms into the desired format. This algebraic manipulation is a fundamental technique in solving equations and is crucial for applying various solution methods.
Performing these operations, we get:
x² + 5x - 8 - 4x - 4 = 0
Combining like terms, we simplify the equation to:
x² + x - 12 = 0
This equation, x² + x - 12 = 0, is now in the standard quadratic form, ax² + bx + c = 0, where a = 1, b = 1, and c = -12. This form allows us to readily apply various methods for solving quadratic equations, including factoring, the quadratic formula, and completing the square. With the equation now in this standard format, we are well-prepared to proceed with finding its solutions.
Now that we have the quadratic equation in its standard form, x² + x - 12 = 0, we can proceed with verifying each of the given options to determine which ones are solutions. This involves substituting each value for 'x' in the equation and checking if the resulting equation holds true. If the left-hand side (LHS) of the equation equals the right-hand side (RHS) after the substitution, then the value is a solution. This methodical approach ensures that we accurately identify the roots of the equation from the provided choices.
A. x = -4
Substituting x = -4 into the equation x² + x - 12 = 0, we get:
(-4)² + (-4) - 12 = 16 - 4 - 12 = 0
Since the equation holds true, x = -4 is a solution.
B. x = -3
Substituting x = -3 into the equation x² + x - 12 = 0, we get:
(-3)² + (-3) - 12 = 9 - 3 - 12 = -6
Since the equation does not hold true, x = -3 is not a solution.
C. x = 4
Substituting x = 4 into the equation x² + x - 12 = 0, we get:
(4)² + (4) - 12 = 16 + 4 - 12 = 8
Since the equation does not hold true, x = 4 is not a solution.
D. x = 5
Substituting x = 5 into the equation x² + x - 12 = 0, we get:
(5)² + (5) - 12 = 25 + 5 - 12 = 18
Since the equation does not hold true, x = 5 is not a solution.
E. x = 3
Substituting x = 3 into the equation x² + x - 12 = 0, we get:
(3)² + (3) - 12 = 9 + 3 - 12 = 0
Since the equation holds true, x = 3 is a solution.
F. x = -2
Substituting x = -2 into the equation x² + x - 12 = 0, we get:
(-2)² + (-2) - 12 = 4 - 2 - 12 = -10
Since the equation does not hold true, x = -2 is not a solution.
While we successfully identified the solutions by substituting the given options, let's explore an alternative method for solving quadratic equations: factoring. Factoring involves expressing the quadratic expression as a product of two linear factors. This method is particularly efficient when dealing with equations that can be easily factored. By factoring the quadratic expression, we can directly determine the values of 'x' that make the equation equal to zero.
Our quadratic equation in standard form is x² + x - 12 = 0. To factor this equation, we need to find two numbers that multiply to -12 (the constant term) and add up to 1 (the coefficient of the 'x' term). After careful consideration, we can identify these numbers as 4 and -3. These numbers satisfy both conditions, as 4 * -3 = -12 and 4 + (-3) = 1.
Using these numbers, we can rewrite the quadratic equation as:
(x + 4)(x - 3) = 0
This factored form of the equation represents the product of two linear expressions. For the product to be zero, at least one of the factors must be zero. This principle allows us to set each factor equal to zero and solve for 'x'. By solving these linear equations, we obtain the roots of the quadratic equation.
Setting each factor equal to zero, we get:
x + 4 = 0 or x - 3 = 0
Solving for 'x' in each equation, we find:
x = -4 or x = 3
These solutions, x = -4 and x = 3, align perfectly with the solutions we identified earlier through substitution. This consistency reinforces the accuracy of our findings and demonstrates the effectiveness of the factoring method. By factoring the quadratic equation, we were able to directly determine its roots, providing an alternative approach to solving the problem.
In conclusion, after meticulously substituting each option into the equation x² + 5x - 8 = 4x + 4, and also by employing the factoring method, we have identified the solutions. The values that satisfy the equation are A. -4 and E. 3. These values, when substituted for 'x' in the equation, make the left-hand side equal to the right-hand side, confirming their validity as solutions.
Throughout this exploration, we've not only found the solutions but also reinforced our understanding of quadratic equations and various problem-solving techniques. We began by transforming the equation into its standard form, a crucial step for applying various solution methods. Then, we systematically substituted each option to verify whether it satisfied the equation. Additionally, we demonstrated the factoring method as an alternative approach, further solidifying our grasp of quadratic equation solutions.
This exercise highlights the importance of algebraic manipulation in simplifying equations and the power of different solution methods. Whether through substitution or factoring, the key lies in understanding the underlying principles and applying them effectively. The solutions we've found represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graph. These points hold significant meaning in various applications, further emphasizing the importance of mastering quadratic equations.
Understanding quadratic equations is a cornerstone of mathematical literacy, with far-reaching implications in various fields. By delving into the process of solving these equations, we equip ourselves with valuable problem-solving skills and a deeper appreciation for the beauty and power of mathematics.