Solving 3x² = -12x - 15 A Step-by-Step Guide
Understanding Quadratic Equations
Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is two. A standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. These equations appear frequently in various fields, including physics, engineering, economics, and computer science. The solutions to a quadratic equation are called roots or zeros, and they represent the values of the variable that satisfy the equation. A quadratic equation can have two real roots, one real root (a repeated root), or two complex roots. The nature of the roots is determined by the discriminant, which we will discuss later in this article.
Methods for Solving Quadratic Equations
There are several methods to solve quadratic equations, each with its advantages and applicability based on the equation's form:
- Factoring: This method involves rewriting the quadratic equation as a product of two binomials. It is efficient when the equation can be easily factored.
- Completing the Square: This technique transforms the quadratic equation into a perfect square trinomial, making it easier to solve. It is a versatile method that works for all quadratic equations.
- Quadratic Formula: The quadratic formula is a general solution that can be applied to any quadratic equation. It provides the roots directly using the coefficients a, b, and c. The formula is given by: x = (-b ± √(b² - 4ac)) / 2a
In this article, we will primarily use the quadratic formula to solve the equation 3x² = -12x - 15, as it is a robust method that guarantees a solution.
Solving 3x² = -12x - 15 Using the Quadratic Formula
To solve the given equation, 3x² = -12x - 15, we will follow these steps:
Step 1: Rewrite the Equation in Standard Form
Before applying the quadratic formula, we need to rewrite the equation in the standard form ax² + bx + c = 0. This involves moving all terms to one side of the equation. Starting with 3x² = -12x - 15, we add 12x and 15 to both sides:
3x² + 12x + 15 = 0
Now, the equation is in the standard form, with a = 3, b = 12, and c = 15.
Step 2: Simplify the Equation (Optional)
Simplifying the equation can make the coefficients smaller and the quadratic formula easier to apply. We observe that all coefficients (3, 12, and 15) are divisible by 3. Dividing the entire equation by 3, we get:
x² + 4x + 5 = 0
This simplified equation has a = 1, b = 4, and c = 5, which are smaller numbers and will simplify our calculations.
Step 3: Apply the Quadratic Formula
The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a. We will substitute the values of a, b, and c from our simplified equation (x² + 4x + 5 = 0) into the formula. Here, a = 1, b = 4, and c = 5. Plugging these values into the formula, we get:
x = (-4 ± √(4² - 4 * 1 * 5)) / (2 * 1)
Step 4: Simplify the Expression
Next, we simplify the expression step by step:
x = (-4 ± √(16 - 20)) / 2 x = (-4 ± √(-4)) / 2
Step 5: Deal with the Imaginary Unit
Since we have a negative number under the square root, we introduce the imaginary unit i, where i² = -1. We can rewrite √(-4) as √(4 * -1) = √(4) * √(-1) = 2i. So, our expression becomes:
x = (-4 ± 2i) / 2
Step 6: Final Simplification
Finally, we divide both the real and imaginary parts by 2:
x = -4/2 ± 2i/2 x = -2 ± i
Thus, the solutions to the equation 3x² = -12x - 15 are x = -2 + i and x = -2 - i.
Analyzing the Roots and the Discriminant
The roots we found, x = -2 + i and x = -2 - i, are complex conjugates. This result is not surprising when we consider the discriminant of the quadratic equation. The discriminant, denoted by Δ, is the part of the quadratic formula under the square root: Δ = b² - 4ac. The discriminant helps determine the nature of the roots:
- If Δ > 0, the equation has two distinct real roots.
- If Δ = 0, the equation has one real root (a repeated root).
- If Δ < 0, the equation has two complex conjugate roots.
In our case, for the simplified equation x² + 4x + 5 = 0, the discriminant is:
Δ = 4² - 4 * 1 * 5 = 16 - 20 = -4
Since Δ < 0, we expect to find two complex conjugate roots, which is exactly what we found.
Alternative Method: Completing the Square
Completing the square is another powerful method to solve quadratic equations. It involves manipulating the equation to form a perfect square trinomial. Let’s apply this method to our simplified equation, x² + 4x + 5 = 0.
Step 1: Move the Constant Term to the Other Side
x² + 4x = -5
Step 2: Complete the Square
To complete the square, we take half of the coefficient of the x term (which is 4), square it (which is (4/2)² = 4), and add it to both sides of the equation:
x² + 4x + 4 = -5 + 4
Step 3: Rewrite as a Perfect Square
The left side is now a perfect square trinomial, which can be written as:
(x + 2)² = -1
Step 4: Take the Square Root of Both Sides
√((x + 2)²) = √(-1)
x + 2 = ± i
Step 5: Solve for x
x = -2 ± i
Again, we arrive at the same solutions, x = -2 + i and x = -2 - i, confirming the accuracy of our method.
Common Mistakes and How to Avoid Them
When solving quadratic equations, several common mistakes can occur. Recognizing and avoiding these mistakes is crucial for obtaining the correct solutions.
1. Incorrectly Applying the Quadratic Formula
A common mistake is misidentifying the coefficients a, b, and c or incorrectly substituting them into the quadratic formula. Always double-check the values and ensure you’re using the correct signs.
2. Errors in Simplifying Expressions
Simplification errors, particularly when dealing with negative signs and square roots, can lead to incorrect answers. Take each step slowly and carefully, and double-check your arithmetic.
3. Forgetting the ± Sign
When taking the square root in methods like completing the square or using the quadratic formula, it’s crucial to remember the ± sign. This accounts for both positive and negative roots.
4. Not Rewriting the Equation in Standard Form
Before applying any method, ensure the equation is in the standard form ax² + bx + c = 0. Failing to do so can lead to incorrect coefficient values and ultimately, wrong solutions.
5. Incorrectly Handling Imaginary Units
When dealing with complex roots, remember that i² = -1. Errors in handling imaginary units can lead to incorrect complex solutions.
Conclusion
In summary, we have successfully solved the quadratic equation 3x² = -12x - 15 using the quadratic formula and the method of completing the square. The solutions are the complex conjugates x = -2 + i and x = -2 - i. Understanding the concepts behind quadratic equations, including the discriminant and different solution methods, is crucial for mastering algebra and its applications. By following the step-by-step guides and avoiding common mistakes, you can confidently solve quadratic equations of various forms. This knowledge is not only valuable for academic pursuits but also for real-world problem-solving in numerous fields.
The correct answer is D. x = -2 ± i.
