Solving $3x^2 - 9x + 4 = 0$ Using The Quadratic Formula
In mathematics, quadratic equations are polynomial equations of the second degree. They have the general form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Solving quadratic equations involves finding the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. One of the most reliable methods for solving quadratic equations is the quadratic formula. This article will delve into how to use the quadratic formula to solve the equation 3x² - 9x + 4 = 0.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool that provides a direct method for finding the solutions of any quadratic equation. It is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Where a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant provides valuable information about the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (a repeated root).
- If b² - 4ac < 0, the equation has two complex roots.
Identifying Coefficients
The first step in using the quadratic formula is to correctly identify the coefficients a, b, and c from the given quadratic equation. For the equation 3x² - 9x + 4 = 0, we can easily identify the coefficients as follows:
- a = 3
- b = -9
- c = 4
Correctly identifying these coefficients is crucial because they will be plugged into the quadratic formula to find the solutions for x. A mistake in identifying the coefficients will lead to incorrect solutions. Therefore, it is always a good practice to double-check these values before proceeding with the calculations. By accurately determining a, b, and c, we set the stage for successfully applying the quadratic formula and solving the quadratic equation.
Applying the Quadratic Formula
Now that we have identified the coefficients a = 3, b = -9, and c = 4, we can substitute these values into the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Substituting the values, we get:
x = (-(-9) ± √((-9)² - 4 * 3 * 4)) / (2 * 3)
This expression can be simplified step by step to find the solutions for x. First, we simplify the terms inside the square root and the denominator:
x = (9 ± √(81 - 48)) / 6
x = (9 ± √33) / 6
Here, we have simplified the expression to a point where we can see that there will be two distinct real roots because the discriminant (33) is positive. The next step involves separating the ± sign to find the two possible values for x. By plugging the coefficients into the formula and carefully simplifying, we are on the verge of discovering the solutions to the quadratic equation. Accurate substitution and simplification are key to arriving at the correct answers.
Simplifying the Expression
Having reached the expression x = (9 ± √33) / 6, we now need to simplify it to find the two distinct roots of the quadratic equation. The ± sign indicates that there are two possible solutions: one where we add √33 and one where we subtract √33. We can write these two solutions as:
x₁ = (9 + √33) / 6
x₂ = (9 - √33) / 6
These solutions can be further simplified by dividing each term in the numerator by the denominator, if possible. In this case, we can express the solutions as fractions. The exact values of x are given by these expressions, which involve the square root of 33. If we need approximate decimal values, we can use a calculator to find the square root of 33 and then perform the calculations.
The simplified expressions for x₁ and x₂ represent the two values of x that satisfy the original quadratic equation 3x² - 9x + 4 = 0. These roots are crucial for understanding the behavior of the quadratic function and can be used in various applications, such as graphing the parabola represented by the equation or solving related problems in physics and engineering.
Finding the Solutions
To find the specific solutions, we calculate the two values of x using the simplified expressions:
x₁ = (9 + √33) / 6
x₂ = (9 - √33) / 6
We can use a calculator to approximate the square root of 33, which is approximately 5.745. Substituting this value, we get:
x₁ ≈ (9 + 5.745) / 6 ≈ 14.745 / 6 ≈ 2.4575
x₂ ≈ (9 - 5.745) / 6 ≈ 3.255 / 6 ≈ 0.5425
Therefore, the two solutions for x are approximately 2.4575 and 0.5425. These solutions represent the points where the parabola described by the equation 3x² - 9x + 4 = 0 intersects the x-axis. It is always a good practice to check these solutions by plugging them back into the original equation to ensure they satisfy it. This step helps verify the accuracy of the calculations and confirms that the correct roots have been found.
Expressing the Solutions
The solutions for the quadratic equation 3x² - 9x + 4 = 0, obtained using the quadratic formula, are:
x₁ ≈ 2.4575
x₂ ≈ 0.5425
When expressing the solutions, it is important to specify whether exact values or approximate values are being provided. In this case, we have approximate decimal values. If the problem requires exact values, the solutions would be expressed in terms of the square root:
x₁ = (9 + √33) / 6
x₂ = (9 - √33) / 6
The context of the problem and the instructions given will dictate whether to provide exact or approximate solutions. In many practical applications, approximate decimal values are sufficient, but in theoretical mathematics, exact values are often preferred. Regardless of the format, it is essential to clearly and accurately present the solutions to ensure they are correctly interpreted.
Conclusion
In conclusion, the quadratic formula is a powerful and versatile tool for solving quadratic equations of the form ax² + bx + c = 0. By correctly identifying the coefficients a, b, and c, and substituting them into the formula x = (-b ± √(b² - 4ac)) / (2a), we can find the solutions for x. For the specific equation 3x² - 9x + 4 = 0, the solutions were found to be approximately 2.4575 and 0.5425. These solutions represent the values of x that satisfy the equation and can be used in various mathematical and practical contexts.
The quadratic formula is not only a means to an end but also a fundamental concept in algebra that demonstrates the relationship between the coefficients of a quadratic equation and its roots. Mastering the use of the quadratic formula is essential for anyone studying mathematics, physics, or engineering, as it provides a reliable method for solving a wide range of problems involving quadratic equations.
