Solving 2x² + 7x = -5 A Step-by-Step Guide
In the realm of mathematics, solving quadratic equations is a fundamental skill. Quadratic equations, characterized by the presence of a squared term, appear in various real-world applications, ranging from physics and engineering to economics and finance. This article delves into the intricacies of solving the specific quadratic equation 2x² + 7x = -5, providing a step-by-step guide that caters to both beginners and those seeking to refresh their understanding. We will explore the standard form of a quadratic equation, various methods for finding solutions (also known as roots), and practical tips for ensuring accuracy. By the end of this guide, you will be equipped with the knowledge and confidence to tackle similar quadratic equations with ease.
Understanding Quadratic Equations
Before diving into the solution of 2x² + 7x = -5, it's crucial to grasp the fundamental concept of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The standard form of a quadratic equation is expressed as:
ax² + bx + c = 0
where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The coefficient 'a' cannot be zero, as this would reduce the equation to a linear form. The term 'ax²' is the quadratic term, 'bx' is the linear term, and 'c' is the constant term. Solving a quadratic equation involves finding the values of 'x' that satisfy the equation, which are also known as the roots or solutions of the equation. These roots represent the points where the parabola, the graphical representation of the quadratic equation, intersects the x-axis. Quadratic equations can have two distinct real roots, one repeated real root, or two complex roots. The nature of the roots depends on the discriminant, which is a key component of the quadratic formula.
Transforming the Equation into Standard Form
Our initial equation is 2x² + 7x = -5. To effectively solve it, we must first transform it into the standard form ax² + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. In this case, we need to add 5 to both sides of the equation:
2x² + 7x + 5 = 0
Now, the equation is in the standard form, where a = 2, b = 7, and c = 5. This transformation is a critical first step because many solution methods, such as factoring and the quadratic formula, are designed to work with equations in standard form. Identifying the values of a, b, and c is crucial for applying these methods correctly. The standard form also allows us to easily calculate the discriminant, which helps determine the nature of the roots before we even attempt to solve the equation. This preparatory step ensures that we approach the solution process in a structured and organized manner.
Methods for Solving Quadratic Equations
There are several methods available for solving quadratic equations, each with its own strengths and weaknesses. The most common methods include:
- Factoring: This method involves expressing the quadratic expression as a product of two linear factors. It is often the quickest method when applicable, but it is not always straightforward, especially when the coefficients are large or the roots are not rational.
- Quadratic Formula: This is a universal method that can solve any quadratic equation, regardless of the nature of its roots. It is derived by completing the square on the standard form of the quadratic equation and provides a direct formula for calculating the roots.
- Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial, which can then be easily factored. It is a powerful method but can be more time-consuming than factoring or using the quadratic formula.
Factoring the Quadratic Equation
Let's attempt to solve 2x² + 7x + 5 = 0 by factoring. Factoring involves finding two binomials that multiply together to give the quadratic expression. We need to find two numbers that multiply to (2)(5) = 10 and add up to 7. These numbers are 2 and 5. We can rewrite the middle term (7x) as the sum of 2x and 5x:
2x² + 2x + 5x + 5 = 0
Now, we can factor by grouping. Group the first two terms and the last two terms:
(2x² + 2x) + (5x + 5) = 0
Factor out the greatest common factor from each group:
2x(x + 1) + 5(x + 1) = 0
Notice that (x + 1) is a common factor in both terms. Factor it out:
(2x + 5)(x + 1) = 0
Now, we have factored the quadratic equation into the product of two binomials. To find the solutions, we set each factor equal to zero:
2x + 5 = 0 or x + 1 = 0
Solving each equation for x gives us:
x = -5/2 or x = -1
Therefore, the roots of the equation 2x² + 7x + 5 = 0 are x = -5/2 and x = -1.
Using the Quadratic Formula
The quadratic formula is a powerful tool for solving any quadratic equation of the form ax² + bx + c = 0. The formula is given by:
x = [-b ± √(b² - 4ac)] / (2a)
In our equation, 2x² + 7x + 5 = 0, we have a = 2, b = 7, and c = 5. Plugging these values into the quadratic formula, we get:
x = [-7 ± √(7² - 4(2)(5))] / (2(2))
x = [-7 ± √(49 - 40)] / 4
x = [-7 ± √9] / 4
x = [-7 ± 3] / 4
This gives us two possible solutions:
x₁ = (-7 + 3) / 4 = -4 / 4 = -1
x₂ = (-7 - 3) / 4 = -10 / 4 = -5/2
As we can see, the quadratic formula yields the same solutions as factoring: x = -1 and x = -5/2. The quadratic formula is particularly useful when the equation is difficult or impossible to factor, or when the roots are irrational or complex.
Completing the Square Method
Completing the square is another method for solving quadratic equations. It involves transforming the equation into a perfect square trinomial. Let's apply this method to 2x² + 7x + 5 = 0. First, divide the entire equation by the coefficient of x², which is 2:
x² + (7/2)x + 5/2 = 0
Next, move the constant term to the right side of the equation:
x² + (7/2)x = -5/2
To complete the square, we need to add (b/2a)² to both sides of the equation, where b is the coefficient of x and a is the coefficient of x² (which is 1 in this case). So, we add ((7/2)/2)² = (7/4)² = 49/16 to both sides:
x² + (7/2)x + 49/16 = -5/2 + 49/16
Now, the left side is a perfect square trinomial:
(x + 7/4)² = -40/16 + 49/16
(x + 7/4)² = 9/16
Take the square root of both sides:
x + 7/4 = ±√(9/16)
x + 7/4 = ±3/4
Solve for x:
x = -7/4 ± 3/4
This gives us two solutions:
x₁ = (-7/4) + (3/4) = -4/4 = -1
x₂ = (-7/4) - (3/4) = -10/4 = -5/2
Again, we obtain the same solutions: x = -1 and x = -5/2. Completing the square is a valuable method, especially for understanding the derivation of the quadratic formula and for solving equations where factoring is not straightforward.
Verifying the Solutions
After finding the solutions to a quadratic equation, it's essential to verify them. This ensures that our calculations are correct and that the solutions satisfy the original equation. To verify, we substitute each solution back into the original equation and check if the equation holds true. Let's verify our solutions, x = -1 and x = -5/2, for the equation 2x² + 7x = -5.
Verifying x = -1
Substitute x = -1 into the equation:
2(-1)² + 7(-1) = -5
2(1) - 7 = -5
2 - 7 = -5
-5 = -5
The equation holds true for x = -1, so it is a valid solution.
Verifying x = -5/2
Substitute x = -5/2 into the equation:
2(-5/2)² + 7(-5/2) = -5
2(25/4) - 35/2 = -5
25/2 - 35/2 = -5
-10/2 = -5
-5 = -5
The equation also holds true for x = -5/2, confirming that it is a valid solution. By verifying our solutions, we can be confident in the accuracy of our work and the correctness of our answers.
Conclusion
Solving the quadratic equation 2x² + 7x = -5 has provided a comprehensive exploration of various methods, including factoring, the quadratic formula, and completing the square. Each method offers a unique approach, and understanding their strengths and weaknesses allows for a flexible and efficient problem-solving strategy. We successfully found the solutions x = -1 and x = -5/2, and we verified their correctness by substituting them back into the original equation. Mastering these techniques is crucial for success in algebra and beyond, as quadratic equations appear in numerous mathematical and real-world contexts. This guide serves as a valuable resource for students, educators, and anyone seeking to enhance their understanding of quadratic equations and their solutions. By practicing and applying these methods, you can confidently tackle any quadratic equation that comes your way.
