Solving Quadratic Equations Step By Step Guide

Quadratic equations are a fundamental part of algebra, and mastering their solutions is crucial for various mathematical and scientific applications. This comprehensive guide delves into solving ten different quadratic equations, providing a step-by-step approach for each. We will explore various techniques, including isolating the variable and applying the square root property. This detailed explanation aims to enhance your understanding and proficiency in solving quadratic equations, making this essential skill more accessible and manageable.

1) x² - 100 = 0

To solve the quadratic equation x² - 100 = 0, our primary goal is to isolate the x² term on one side of the equation. This involves adding 100 to both sides of the equation. By doing so, we maintain the equation's balance while moving the constant term to the right side. The equation then transforms into x² = 100, which is a much simpler form to work with. Now that we have x² isolated, the next crucial step is to eliminate the square. To achieve this, we apply the square root operation to both sides of the equation. Remember, when we take the square root of a number, we must consider both the positive and negative roots, as both will satisfy the original equation. Thus, we have x = ±√100. The square root of 100 is 10, so the solutions are x = ±10. This means that both 10 and -10, when squared, will result in 100, satisfying the initial equation. Therefore, the solutions to the equation x² - 100 = 0 are x = 10 and x = -10. This process of isolating the variable and then using the square root property is a fundamental technique in solving many quadratic equations.

2) 2x² - 50 = 0

In tackling the quadratic equation 2x² - 50 = 0, our initial aim is to isolate the term containing x². This is a common strategy in solving quadratic equations, allowing us to simplify the equation step by step. We begin by adding 50 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation 2x² = 50. The next step is to further isolate x² by dividing both sides of the equation by 2. This division removes the coefficient of x², leading to a more straightforward equation: x² = 25. With x² now isolated, we can proceed to the next phase, which involves eliminating the square. To do this, we apply the square root operation to both sides of the equation. When taking the square root, it's crucial to remember that we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive result. Thus, we have x = ±√25. The square root of 25 is 5, so the solutions are x = ±5. This means the solutions to the equation 2x² - 50 = 0 are x = 5 and x = -5. These values, when substituted back into the original equation, will satisfy the equation, confirming their validity.

3) x² - 2 = 14

To solve the quadratic equation x² - 2 = 14, the first step involves isolating the x² term on one side of the equation. This is a fundamental technique in solving quadratic equations, as it simplifies the equation and makes it easier to find the solutions. We begin by adding 2 to both sides of the equation. This operation cancels out the -2 on the left side and moves the constant term to the right side, resulting in the equation x² = 16. Now that we have x² isolated, we can proceed to eliminate the square. This is achieved by taking the square root of both sides of the equation. When applying the square root, it's essential to remember to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will result in the same positive number. Thus, we have x = ±√16. The square root of 16 is 4, so the solutions are x = ±4. This means that the solutions to the equation x² - 2 = 14 are x = 4 and x = -4. Both these values, when substituted back into the original equation, will satisfy the equation, confirming that they are indeed the solutions. Isolating the x² term and then applying the square root property is a common and effective method for solving quadratic equations of this form.

4) x² - 196 = 0

Solving the quadratic equation x² - 196 = 0 involves a similar approach to the previous examples. Our initial focus is on isolating the x² term on one side of the equation. This is a crucial step in simplifying the equation and making it easier to solve. To achieve this, we add 196 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation x² = 196. Now that we have x² isolated, we can eliminate the square by taking the square root of both sides of the equation. When taking the square root, it's vital to remember that we must consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive number. Thus, we have x = ±√196. The square root of 196 is 14, so the solutions are x = ±14. This means that the solutions to the equation x² - 196 = 0 are x = 14 and x = -14. When either 14 or -14 is squared, the result is 196, which satisfies the original equation. Therefore, these are the correct solutions. This method of isolating the x² term and then applying the square root property is a fundamental technique in solving quadratic equations of this type.

5) x² - 27 = -2

To solve the quadratic equation x² - 27 = -2, we begin by isolating the x² term. This is a standard procedure in solving quadratic equations, as it simplifies the equation and brings us closer to finding the solutions. We start by adding 27 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation x² = 25. With x² now isolated, our next step is to eliminate the square. We achieve this by taking the square root of both sides of the equation. It's crucial to remember that when taking the square root, we must consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive number. Thus, we have x = ±√25. The square root of 25 is 5, so the solutions are x = ±5. This means that the solutions to the equation x² - 27 = -2 are x = 5 and x = -5. When either 5 or -5 is squared, the result is 25, which satisfies the transformed equation and consequently the original equation. Therefore, these are the correct solutions. Isolating the x² term and then applying the square root property is a common and effective method for solving quadratic equations of this form.

6) 4x² - 49 = 0

In solving the quadratic equation 4x² - 49 = 0, our initial step is to isolate the term containing x². This is a fundamental strategy in solving quadratic equations, allowing us to simplify the equation progressively. We begin by adding 49 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation 4x² = 49. The next step is to further isolate x² by dividing both sides of the equation by 4. This division removes the coefficient of x², leading to the equation x² = 49/4. With x² now isolated, we can proceed to the next phase, which involves eliminating the square. To do this, we apply the square root operation to both sides of the equation. When taking the square root, it's crucial to remember that we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive result. Thus, we have x = ±√(49/4). The square root of 49/4 is ±7/2, so the solutions are x = ±7/2. This means the solutions to the equation 4x² - 49 = 0 are x = 7/2 and x = -7/2. These values, when substituted back into the original equation, will satisfy the equation, confirming their validity.

7) 3x² - 24 = 0

To solve the quadratic equation 3x² - 24 = 0, our initial goal is to isolate the term containing x². This is a standard approach in solving quadratic equations, as it simplifies the equation step by step. We begin by adding 24 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation 3x² = 24. The next step is to further isolate x² by dividing both sides of the equation by 3. This division removes the coefficient of x², leading to a more straightforward equation: x² = 8. With x² now isolated, we can proceed to the next phase, which involves eliminating the square. To do this, we apply the square root operation to both sides of the equation. When taking the square root, it's crucial to remember that we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive result. Thus, we have x = ±√8. The square root of 8 can be simplified to 2√2, so the solutions are x = ±2√2. This means the solutions to the equation 3x² - 24 = 0 are x = 2√2 and x = -2√2. These values, when squared and substituted back into the original equation, will satisfy the equation, confirming their validity.

8) 5x² - 3 = x² + 78

In tackling the quadratic equation 5x² - 3 = x² + 78, the initial step involves rearranging the terms to group like terms together. This is a crucial strategy in simplifying the equation and making it easier to solve. We begin by subtracting x² from both sides of the equation. This action moves the x² term from the right side to the left side, resulting in the equation 4x² - 3 = 78. The next step is to isolate the term containing x². We do this by adding 3 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation 4x² = 81. Now, we further isolate x² by dividing both sides of the equation by 4. This division removes the coefficient of x², leading to a more straightforward equation: x² = 81/4. With x² now isolated, we can proceed to the next phase, which involves eliminating the square. To do this, we apply the square root operation to both sides of the equation. When taking the square root, it's crucial to remember that we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive result. Thus, we have x = ±√(81/4). The square root of 81/4 is ±9/2, so the solutions are x = ±9/2. This means the solutions to the equation 5x² - 3 = x² + 78 are x = 9/2 and x = -9/2. These values, when substituted back into the original equation, will satisfy the equation, confirming their validity.

9) 16x² - 121 = 0

Solving the quadratic equation 16x² - 121 = 0 requires a systematic approach. Our first goal is to isolate the term containing x². This is a common technique in solving quadratic equations, allowing us to simplify the equation step by step. We begin by adding 121 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation 16x² = 121. The next step is to further isolate x² by dividing both sides of the equation by 16. This division removes the coefficient of x², leading to a more straightforward equation: x² = 121/16. With x² now isolated, we can proceed to the next phase, which involves eliminating the square. To do this, we apply the square root operation to both sides of the equation. When taking the square root, it's crucial to remember that we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive result. Thus, we have x = ±√(121/16). The square root of 121/16 is ±11/4, so the solutions are x = ±11/4. This means the solutions to the equation 16x² - 121 = 0 are x = 11/4 and x = -11/4. These values, when substituted back into the original equation, will satisfy the equation, confirming their validity.

10) 5(x² - 3) = 0

To solve the quadratic equation 5(x² - 3) = 0, the first step involves simplifying the equation by distributing the 5 across the terms inside the parentheses. This is a crucial step in making the equation easier to solve. By distributing the 5, we obtain the equation 5x² - 15 = 0. Now, our goal is to isolate the term containing x². This is a standard technique in solving quadratic equations. We begin by adding 15 to both sides of the equation. This action moves the constant term to the right side, resulting in the equation 5x² = 15. The next step is to further isolate x² by dividing both sides of the equation by 5. This division removes the coefficient of x², leading to a more straightforward equation: x² = 3. With x² now isolated, we can proceed to the next phase, which involves eliminating the square. To do this, we apply the square root operation to both sides of the equation. When taking the square root, it's crucial to remember that we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive result. Thus, we have x = ±√3. The square root of 3 is an irrational number, so we leave it in radical form. The solutions are x = √3 and x = -√3. This means the solutions to the equation 5(x² - 3) = 0 are x = √3 and x = -√3. These values, when squared and substituted back into the original equation, will satisfy the equation, confirming their validity.

Conclusion

In conclusion, solving quadratic equations is a fundamental skill in mathematics. This guide has provided a detailed, step-by-step approach to solving ten different quadratic equations. By isolating the x² term and applying the square root property, we have successfully found the solutions for each equation. Remember to always consider both positive and negative roots when taking the square root. Mastering these techniques will significantly enhance your ability to solve quadratic equations and tackle more complex mathematical problems.