Solving Quadratic Equations Using The Quadratic Formula A Step-by-Step Guide

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Solving quadratic equations is a fundamental skill in algebra, and the quadratic formula is a powerful tool for finding solutions, especially when factoring or other methods are not straightforward. This article will guide you through the process of using the quadratic formula to solve the equation 5x2=2x+45x^2 = 2x + 4. We'll break down each step, ensuring you understand how to apply the formula effectively. Let's dive in and master this essential technique!

Understanding the Quadratic Formula

Before we tackle the specific equation, let's review the quadratic formula itself. The quadratic formula is used to find the solutions (also called roots or zeros) of any quadratic equation in the standard form of ax2+bx+c=0ax^2 + bx + c = 0. The formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Where:

  • xx represents the solutions to the equation.
  • aa, bb, and cc are coefficients derived from the quadratic equation in its standard form.

Key Components of the Quadratic Formula

  1. Standard Form: Before applying the formula, it's crucial to express the quadratic equation in its standard form (ax2+bx+c=0ax^2 + bx + c = 0). This ensures you correctly identify the coefficients aa, bb, and cc.
  2. Coefficients a, b, and c: These are the numerical coefficients of the quadratic equation. 'aa' is the coefficient of the x2x^2 term, 'bb' is the coefficient of the xx term, and 'cc' is the constant term.
  3. The Discriminant: The expression inside the square root, b2−4acb^2 - 4ac, is known as the discriminant. The discriminant plays a vital role in determining the nature of the solutions:
    • If b2−4ac>0b^2 - 4ac > 0, the equation has two distinct real solutions.
    • If b2−4ac=0b^2 - 4ac = 0, the equation has one real solution (a repeated root).
    • If b2−4ac<0b^2 - 4ac < 0, the equation has two complex solutions.

Why Use the Quadratic Formula?

The quadratic formula is a versatile method for solving quadratic equations because it works for all quadratic equations, regardless of whether they can be factored easily. Factoring is a quick method when it works, but many quadratic equations are difficult or impossible to factor. Other methods, such as completing the square, can be used, but the quadratic formula provides a direct and reliable way to find the solutions. The quadratic formula is especially useful when the roots are irrational or complex, as these solutions are not readily obtainable through factoring.

Understanding these basics is crucial before we apply the quadratic formula to our given equation. With a firm grasp of these concepts, we can confidently solve any quadratic equation using this powerful tool. The quadratic formula ensures that we can always find the solutions, regardless of the complexity of the equation. As a result, it is an indispensable tool in algebra.

Step-by-Step Solution for 5x2=2x+45x^2 = 2x + 4

Now, let's apply the quadratic formula to solve the given equation, 5x2=2x+45x^2 = 2x + 4. We'll follow a step-by-step approach to ensure clarity and accuracy. Solving the equation involves rewriting it in standard form, identifying the coefficients, substituting them into the quadratic formula, and simplifying the result to find the solutions.

1. Rewrite the Equation in Standard Form

The first step is to rewrite the given equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. For the equation 5x2=2x+45x^2 = 2x + 4, we subtract 2x2x and 44 from both sides:

5x2−2x−4=05x^2 - 2x - 4 = 0

Now, the equation is in standard form, and we can easily identify the coefficients.

2. Identify the Coefficients a, b, and c

In the standard form equation 5x2−2x−4=05x^2 - 2x - 4 = 0, we can now identify the coefficients:

  • a=5a = 5 (the coefficient of x2x^2)
  • b=−2b = -2 (the coefficient of xx)
  • c=−4c = -4 (the constant term)

Correctly identifying these coefficients is crucial because they will be substituted into the quadratic formula.

3. Substitute the Coefficients into the Quadratic Formula

Next, we substitute the values of aa, bb, and cc into the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting a=5a = 5, b=−2b = -2, and c=−4c = -4, we get:

x=−(−2)±(−2)2−4(5)(−4)2(5)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-4)}}{2(5)}

4. Simplify the Expression

Now, we simplify the expression step by step. First, simplify the terms inside the square root and the denominator:

x=2±4+8010x = \frac{2 \pm \sqrt{4 + 80}}{10}

x=2±8410x = \frac{2 \pm \sqrt{84}}{10}

Next, simplify the square root. Since 84=4imes2184 = 4 imes 21, we can simplify 84\sqrt{84} as follows:

84=4imes21=4imes21=221\sqrt{84} = \sqrt{4 imes 21} = \sqrt{4} imes \sqrt{21} = 2\sqrt{21}

Substitute this back into the equation:

x=2±22110x = \frac{2 \pm 2\sqrt{21}}{10}

5. Further Simplification

Finally, we simplify the entire expression by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

x=2(1±21)10x = \frac{2(1 \pm \sqrt{21})}{10}

x=1±215x = \frac{1 \pm \sqrt{21}}{5}

Thus, the solutions are:

x=1+215andx=1−215x = \frac{1 + \sqrt{21}}{5} \quad \text{and} \quad x = \frac{1 - \sqrt{21}}{5}

By following these steps, we have successfully applied the quadratic formula to solve the equation 5x2=2x+45x^2 = 2x + 4. This methodical approach ensures accurate results and a clear understanding of the process. The solutions are exact values, expressed using radicals, which is a common outcome when using the quadratic formula for equations that do not factor neatly. Now, let's summarize our findings and present the solution set.

Expressing the Solution Set

After applying the quadratic formula and simplifying the results, we found two distinct real solutions for the equation 5x2=2x+45x^2 = 2x + 4. These solutions are:

x=1+215andx=1−215x = \frac{1 + \sqrt{21}}{5} \quad \text{and} \quad x = \frac{1 - \sqrt{21}}{5}

To express the solution set, we use set notation, which is a concise way to represent all solutions in a single expression. The solution set is enclosed in curly braces, with the solutions separated by a comma.

Solution Set Notation

The solution set for the equation 5x2=2x+45x^2 = 2x + 4 is:

{1+215,1−215}\left\{\frac{1 + \sqrt{21}}{5}, \frac{1 - \sqrt{21}}{5}\right\}

This notation clearly presents all the solutions we found using the quadratic formula. It is essential to use exact answers, especially when dealing with irrational numbers like radicals, to maintain precision. Approximating the solutions as decimals may lose accuracy and is generally not preferred in mathematical contexts unless specifically requested.

Importance of Exact Answers

In mathematics, especially in higher-level studies, providing exact answers is crucial. Exact answers preserve the true value of the solution without rounding errors. In our case, leaving the solutions in radical form ensures that we maintain the exact values. If we were to approximate 21\sqrt{21} as a decimal, we would introduce a degree of error. For many applications, such as further calculations or theoretical work, these errors can compound and lead to inaccurate results.

Practical Implications

The solutions to quadratic equations have many practical applications in various fields, including physics, engineering, and economics. For example, in physics, quadratic equations can model projectile motion, and the solutions represent the times at which the projectile reaches certain heights. In engineering, quadratic equations can be used to design structures and optimize processes. In economics, they can model cost and revenue functions. Thus, the ability to accurately solve quadratic equations is a valuable skill.

In summary, the solution set for the equation 5x2=2x+45x^2 = 2x + 4, found using the quadratic formula, is expressed as {1+215,1−215}\left\{\frac{1 + \sqrt{21}}{5}, \frac{1 - \sqrt{21}}{5}\right\}. This exact representation ensures accuracy and is consistent with mathematical conventions. Understanding how to find and express solutions in this way is crucial for further studies and applications in various fields. The quadratic formula is an indispensable tool for these applications.

Conclusion

In this article, we've walked through the process of solving the quadratic equation 5x2=2x+45x^2 = 2x + 4 using the quadratic formula. We began by understanding the quadratic formula itself and its key components, such as the coefficients aa, bb, and cc, and the discriminant. Then, we meticulously followed the steps to rewrite the equation in standard form, identify the coefficients, substitute them into the formula, simplify the expression, and express the solution set.

Key Takeaways

  1. Standard Form is Essential: Always rewrite the quadratic equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0 before applying the quadratic formula. This ensures you correctly identify the coefficients.
  2. Careful Substitution: Pay close attention when substituting the coefficients into the quadratic formula. A mistake in substitution can lead to incorrect solutions.
  3. Simplify Thoroughly: Simplify the expression step by step, especially the square root term. Look for perfect square factors to reduce the radical.
  4. Exact Answers: Whenever possible, provide exact answers using radicals rather than decimal approximations. This preserves accuracy.
  5. Solution Set Notation: Use set notation to express the solution set, enclosing the solutions in curly braces.

The Power of the Quadratic Formula

The quadratic formula is a powerful tool because it provides a reliable method for solving any quadratic equation. Unlike factoring, which can be challenging or impossible for some equations, the quadratic formula always yields the solutions, whether they are real or complex. This makes it an indispensable tool in algebra and other areas of mathematics.

Practical Application

The ability to solve quadratic equations is not just a theoretical exercise; it has numerous practical applications. From physics to engineering to economics, quadratic equations model real-world phenomena. For instance, they can be used to calculate the trajectory of a projectile, design the optimal shape for a bridge, or model the growth of a population.

Final Thoughts

Mastering the quadratic formula is a crucial step in developing your algebraic skills. By understanding the formula and practicing its application, you can confidently solve a wide range of quadratic equations. Remember to follow the steps methodically, double-check your work, and always strive for exact answers. With these skills, you'll be well-equipped to tackle more advanced mathematical problems and real-world applications. The quadratic formula is a cornerstone of algebra, and its mastery opens doors to deeper mathematical understanding and problem-solving capabilities.