Solving 5x = 6x² - 3 Using The Quadratic Formula A Step-by-Step Guide

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The quadratic formula is a powerful tool for finding the solutions (also called roots or zeros) of any quadratic equation. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The quadratic formula provides a straightforward method to determine the values of x that satisfy this equation, regardless of whether the equation can be easily factored. In this comprehensive guide, we will walk through the steps of using the quadratic formula to solve the equation 5x = 6x² - 3, and we'll also explore the underlying concepts and potential challenges.

When dealing with quadratic equations, the first step is to ensure the equation is in the standard form ax² + bx + c = 0. This form allows us to clearly identify the coefficients a, b, and c, which are essential for applying the quadratic formula. In our given equation, 5x = 6x² - 3, we need to rearrange the terms to achieve the standard form. Subtracting 5x from both sides and rearranging the terms gives us 6x² - 5x - 3 = 0. Now, we can easily see that a = 6, b = -5, and c = -3. Identifying these coefficients correctly is crucial because they will be directly substituted into the quadratic formula. A mistake in identifying these values can lead to incorrect solutions, so it’s important to double-check and ensure accuracy before proceeding further. Once the coefficients are correctly identified, the next step is to substitute them into the quadratic formula, which will be discussed in detail in the following sections.

The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a). This formula is derived from the method of completing the square and provides a general solution for any quadratic equation. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the solutions. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has exactly one real solution (a repeated root). If it is negative, the equation has two complex solutions. Understanding the discriminant can help predict the type of solutions you will obtain even before fully applying the formula. In our equation, 6x² - 5x - 3 = 0, we identified a = 6, b = -5, and c = -3. Now, we substitute these values into the quadratic formula. This gives us x = (-(-5) ± √((-5)² - 4(6)(-3))) / (2(6)). The next step involves simplifying this expression, which requires careful attention to arithmetic and order of operations. By substituting the values correctly and proceeding with simplification, we can find the solutions for x.

Simplifying the expression obtained after substituting the values into the quadratic formula is a crucial step to arrive at the solutions. We have x = (-(-5) ± √((-5)² - 4(6)(-3))) / (2(6)). First, we simplify the terms inside the square root: (-5)² = 25 and 4(6)(-3) = -72. So, the expression becomes x = (5 ± √(25 - (-72))) / 12. Next, we simplify the discriminant: 25 - (-72) = 25 + 72 = 97. Thus, we have x = (5 ± √97) / 12. The square root of 97 cannot be simplified further as 97 is a prime number. Therefore, we have two solutions for x: x = (5 + √97) / 12 and x = (5 - √97) / 12. These are the exact solutions to the quadratic equation 6x² - 5x - 3 = 0. It’s important to note that these solutions are irrational numbers because of the presence of the square root of 97. Unless asked for decimal approximations, it's best to leave the solutions in this exact form. The process of simplification requires careful attention to detail to avoid arithmetic errors, which can significantly alter the final solutions. Once simplified, we can clearly see the two distinct real solutions, as indicated by the positive discriminant.

To effectively solve the quadratic equation 5x = 6x² - 3 using the quadratic formula, it’s essential to follow a structured approach. This ensures accuracy and clarity in each step. Here, we provide a detailed, step-by-step guide to help you navigate the process, making it easier to understand and apply.

Step 1: Rewrite the Equation in Standard Form

The first critical step in solving any quadratic equation is to rewrite it in the standard form, which is ax² + bx + c = 0. This form is essential because it allows us to clearly identify the coefficients a, b, and c, which are necessary for applying the quadratic formula. The given equation is 5x = 6x² - 3. To convert it into the standard form, we need to rearrange the terms so that one side of the equation is equal to zero. We can achieve this by subtracting 5x from both sides of the equation. This gives us 0 = 6x² - 5x - 3. It's conventional to write the equation with the terms in descending order of the exponent, so we rewrite it as 6x² - 5x - 3 = 0. This standard form makes it easy to identify the coefficients a, b, and c. In this case, a is the coefficient of x², which is 6; b is the coefficient of x, which is -5; and c is the constant term, which is -3. Ensuring the equation is in standard form is a fundamental step, and any errors in this step will propagate through the rest of the solution. Therefore, it’s important to double-check that the terms are correctly rearranged and that the coefficients are accurately identified.

Step 2: Identify the Coefficients a, b, and c

Once the equation is in the standard form ax² + bx + c = 0, the next crucial step is to identify the coefficients a, b, and c. These coefficients are the numerical values that define the quadratic equation, and they are essential for the correct application of the quadratic formula. In our equation, 6x² - 5x - 3 = 0, we can directly observe the values of a, b, and c. The coefficient a is the number multiplying x², which in this case is 6. So, a = 6. The coefficient b is the number multiplying x, which is -5. Therefore, b = -5. It’s important to pay attention to the sign of the coefficient; in this case, b is negative. The constant term c is the number that stands alone, without any x term. Here, c is -3. Again, the sign is crucial, so c = -3. Accurately identifying a, b, and c is paramount because these values will be substituted into the quadratic formula. An error in this step will lead to incorrect solutions. It's a good practice to write these values down clearly (a = 6, b = -5, c = -3) to avoid confusion in the subsequent steps. With the coefficients correctly identified, we can proceed to apply the quadratic formula.

Step 3: Apply the Quadratic Formula

With the coefficients a, b, and c identified, the next step is to apply the quadratic formula. The quadratic formula is a general solution for any quadratic equation in the form ax² + bx + c = 0, and it is given by x = (-b ± √(b² - 4ac)) / (2a). This formula provides a direct method for finding the roots of the equation. In our case, we have a = 6, b = -5, and c = -3. We now substitute these values into the formula. This gives us x = (-(-5) ± √((-5)² - 4(6)(-3))) / (2(6)). It's essential to substitute the values carefully, paying close attention to signs and ensuring that each value is placed in the correct position in the formula. The expression now looks complex, but it is a straightforward substitution. The next step involves simplifying this expression, which includes dealing with the negative signs, squaring the term, and performing the multiplication. This careful substitution is a critical step in ensuring the accuracy of the final solutions. With the values correctly substituted, we can proceed to simplify the expression and find the roots of the equation.

Step 4: Simplify the Expression

After substituting the values into the quadratic formula, the next crucial step is to simplify the resulting expression. This involves performing the arithmetic operations in the correct order to arrive at the solutions for x. We have x = (-(-5) ± √((-5)² - 4(6)(-3))) / (2(6)). First, let’s simplify the terms inside the square root and the denominator. The term -(-5) simplifies to 5. Inside the square root, (-5)² is 25, and 4(6)(-3) is -72. So, the expression becomes x = (5 ± √(25 - (-72))) / (2(6)). Next, we simplify the expression inside the square root further: 25 - (-72) is the same as 25 + 72, which equals 97. The denominator 2(6) simplifies to 12. Thus, the expression becomes x = (5 ± √97) / 12. The square root of 97 cannot be simplified further because 97 is a prime number. Therefore, we have arrived at the simplest form of the solutions. The plus-minus (±) sign indicates that there are two solutions: one where we add the square root of 97 to 5 and divide by 12, and another where we subtract the square root of 97 from 5 and divide by 12. This simplification process is vital to obtain the exact solutions for x. Any errors in arithmetic during this step can lead to incorrect answers.

Step 5: State the Solutions

After simplifying the expression obtained from the quadratic formula, the final step is to state the solutions for x. From our previous simplification, we have x = (5 ± √97) / 12. The plus-minus (±) sign indicates that there are two distinct solutions. The first solution is obtained by adding the square root of 97 to 5 and dividing by 12: x₁ = (5 + √97) / 12. The second solution is obtained by subtracting the square root of 97 from 5 and dividing by 12: x₂ = (5 - √97) / 12. These are the exact solutions to the quadratic equation 6x² - 5x - 3 = 0. Since 97 is a prime number, √97 cannot be simplified further, and these solutions are irrational numbers. Unless a decimal approximation is required, it is best to leave the solutions in this exact form. Stating the solutions clearly and accurately is essential to complete the problem. By following all the steps methodically—rewriting the equation in standard form, identifying coefficients, applying the quadratic formula, simplifying the expression, and stating the solutions—we can confidently solve any quadratic equation using this powerful tool.

Solving quadratic equations using the quadratic formula is a straightforward process, but there are several potential pitfalls that can lead to errors. Being aware of these common mistakes and knowing how to avoid them is crucial for ensuring accuracy. Here, we discuss some of the most frequent errors and provide strategies to prevent them.

1. Incorrectly Identifying Coefficients

One of the most common mistakes when using the quadratic formula is incorrectly identifying the coefficients a, b, and c. This typically occurs when the equation is not in the standard form ax² + bx + c = 0, or when the signs of the coefficients are overlooked. To avoid this, always start by rewriting the equation in standard form. For example, if you have an equation like 3x² = 5x - 2, you need to rearrange it to 3x² - 5x + 2 = 0 before identifying the coefficients. Once the equation is in standard form, carefully identify a as the coefficient of x², b as the coefficient of x, and c as the constant term. Pay close attention to the signs. For instance, in the equation 2x² - 7x + 3 = 0, a = 2, b = -7 (not 7), and c = 3. A simple way to double-check is to write down a =, b =, and c = before substituting the values into the formula. This can help you catch errors before they affect the rest of your solution. Incorrectly identifying the coefficients will lead to wrong solutions, so this step requires careful attention and verification.

2. Sign Errors

Sign errors are another frequent pitfall when applying the quadratic formula. The formula involves several negative signs, and mishandling these can lead to incorrect solutions. For example, the formula starts with -b, so if b is already negative, you have -(-b), which becomes positive. Similarly, the term -4ac can be tricky, especially if c is negative. To minimize sign errors, it’s helpful to use parentheses when substituting values into the formula. For instance, if b = -5, a = 2, and c = -3, write the substitution as x = (-(-5) ± √((-5)² - 4(2)(-3))) / (2(2)). This visual cue helps you keep track of the negative signs. Another strategy is to simplify the expression step by step, paying close attention to each sign. For example, break down -4(2)(-3) into -4 * 2 * -3, and handle the multiplication in stages. Double-checking each step and being methodical about handling negative signs can significantly reduce the risk of errors. Sign errors are particularly common but can be largely avoided with careful and deliberate work.

3. Arithmetic Mistakes

Even if the quadratic formula is applied correctly with the right coefficients, arithmetic mistakes during the simplification process can still lead to incorrect answers. Common arithmetic errors include mistakes in squaring numbers, multiplying terms, and adding or subtracting values. For example, miscalculating (-5)² as -25 instead of 25 is a frequent error. Similarly, mistakes can occur when simplifying the discriminant, b² - 4ac. To minimize arithmetic errors, break down the simplification process into smaller, manageable steps. For example, when calculating b² - 4ac, first compute b², then 4ac, and finally perform the subtraction. Write down each intermediate step to keep track of your calculations. Use a calculator to double-check your arithmetic, especially for larger numbers or complex calculations. Another helpful strategy is to review your work after completing the problem, checking each step for potential errors. Arithmetic mistakes are easy to make but can be avoided with careful attention to detail and the use of tools like calculators and step-by-step calculations.

4. Incorrectly Simplifying the Square Root

The square root term in the quadratic formula can sometimes be simplified, and failing to do so is a common mistake. The term under the square root, b² - 4ac, is called the discriminant, and if it is a perfect square, the square root can be simplified to a whole number. If the discriminant is not a perfect square, it may still be possible to simplify the square root by factoring out perfect square factors. For example, if the discriminant is 48, √48 can be simplified to √(16 * 3) = 4√3. Failing to simplify the square root can lead to solutions that are not in their simplest form. To avoid this, always check if the discriminant is a perfect square. If not, look for perfect square factors. Remember to use prime factorization to identify these factors if necessary. If the discriminant is negative, remember that the solutions will be complex numbers involving i, the imaginary unit (√-1). Incorrect simplification of the square root can result in missing the simplest form of the solution, so it's important to check for perfect square factors and simplify accordingly.

5. Forgetting the ± Sign

The quadratic formula gives two solutions because of the ± sign. Forgetting to consider both the positive and negative roots is a common mistake. The ± sign indicates that there are two possible solutions: one where you add the square root term and one where you subtract it. For example, if the formula simplifies to x = (3 ± √5) / 2, you have two solutions: x = (3 + √5) / 2 and x = (3 - √5) / 2. To avoid forgetting the two solutions, make it a habit to write out both solutions separately as soon as you reach the step where the ± sign is present. This will serve as a reminder that there are two answers to consider. Underlining or highlighting the ± sign can also help you remember that it signifies two distinct calculations. Forgetting one of the solutions can lead to an incomplete answer, so it’s important to consciously acknowledge and calculate both possibilities indicated by the ± sign.

In conclusion, solving quadratic equations using the quadratic formula is a fundamental skill in mathematics. By understanding the steps involved—rewriting the equation in standard form, identifying coefficients, applying the formula, simplifying the expression, and stating the solutions—you can confidently tackle any quadratic equation. While the process is straightforward, it is crucial to be aware of common pitfalls such as incorrectly identifying coefficients, making sign errors, arithmetic mistakes, incorrectly simplifying square roots, and forgetting the ± sign. By taking a methodical approach, paying attention to detail, and double-checking your work, you can minimize these errors and ensure accurate results. The quadratic formula is a powerful tool, and with practice and careful application, you can master this essential mathematical skill.