Solving Quadratic Equations Using The Quadratic Formula X^2 - 2x = 15

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In this comprehensive guide, we will delve into the process of solving the quadratic equation $x^2 - 2x = 15$ using the quadratic formula. Quadratic equations, which take the general form of $ax^2 + bx + c = 0$, are fundamental in mathematics and have widespread applications in various fields, including physics, engineering, and economics. The quadratic formula provides a universal method for finding the solutions (also called roots) of any quadratic equation, regardless of its complexity. We'll break down each step, ensuring a clear understanding of how to apply the formula effectively. By the end of this article, you'll be equipped to confidently tackle similar quadratic equation problems.

Understanding Quadratic Equations and the Quadratic Formula

Before we dive into solving the specific equation, let's establish a solid foundation by defining what a quadratic equation is and introducing the quadratic formula. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is given by:

$ax^2 + bx + c = 0$

where 'a', 'b', and 'c' are coefficients, and 'a' is not equal to 0 (if 'a' were 0, the equation would become linear). These coefficients are crucial in determining the solutions of the equation. The quadratic formula is a powerful tool that provides a direct method for finding these solutions. It is expressed as follows:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The formula might seem daunting at first glance, but it's quite straightforward once you understand its components. The symbols '±' indicate that there are typically two solutions: one where you add the square root term and one where you subtract it. The expression inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant plays a critical role in determining the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If it's zero, there is exactly one real solution (a repeated root). And if it's negative, there are two complex solutions. Understanding these fundamental concepts is key to successfully applying the quadratic formula and interpreting the results.

Step 1: Rewrite the Equation in Standard Form

The first crucial step in solving a quadratic equation using the quadratic formula is to rewrite the equation in its standard form: $ax^2 + bx + c = 0$. This form is essential because it allows us to easily identify the coefficients 'a', 'b', and 'c', which are necessary for applying the quadratic formula. Our given equation is $x^2 - 2x = 15$. To transform this into standard form, we need to move all terms to one side of the equation, leaving zero on the other side. In this case, we subtract 15 from both sides of the equation:

$x^2 - 2x - 15 = 0$

Now, the equation is in the standard form. We can clearly see that:

• a = 1 (the coefficient of $x^2$)
• b = -2 (the coefficient of x)
• c = -15 (the constant term)

Identifying these coefficients correctly is paramount, as they will be directly substituted into the quadratic formula. A mistake in identifying any of these coefficients will lead to incorrect solutions. This step might seem simple, but it's a critical foundation for the rest of the solution process. Taking the time to ensure the equation is in standard form and that the coefficients are correctly identified will save you from potential errors later on. This meticulous approach is a hallmark of effective problem-solving in mathematics.

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

Having successfully rewritten the equation in standard form ($x^2 - 2x - 15 = 0$), the next crucial step is to accurately identify the coefficients 'a', 'b', and 'c'. These coefficients are the numerical values that multiply the respective terms in the quadratic equation, and they are essential for the correct application of the quadratic formula. Recall that the standard form of a quadratic equation is $ax^2 + bx + c = 0$. By comparing our equation with the standard form, we can extract the values of 'a', 'b', and 'c'.

In our equation, $x^2 - 2x - 15 = 0$, we can see the following:

• The coefficient of the $x^2$ term is '1'. Even though there's no explicit number written before $x^2$, it's understood that the coefficient is 1. Therefore, a = 1.
• The coefficient of the 'x' term is '-2'. It's crucial to include the negative sign. Therefore, b = -2.
• The constant term is '-15'. Again, the negative sign is important. Therefore, c = -15.

Correctly identifying these coefficients is a critical step because these values will be substituted directly into the quadratic formula. Any error in identifying these values will lead to an incorrect solution. It's a good practice to double-check these values before proceeding to the next step to minimize the chances of making a mistake. This meticulous attention to detail is a key characteristic of successful problem-solving in mathematics. Once we have correctly identified 'a', 'b', and 'c', we can confidently move on to applying the quadratic formula.

Step 3: Apply the Quadratic Formula

With the coefficients 'a', 'b', and 'c' correctly identified, we are now ready to apply the quadratic formula. The quadratic formula, as we discussed earlier, is a powerful tool for finding the solutions to any quadratic equation in the standard form $ax^2 + bx + c = 0$. The formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula might seem complex at first, but it's a straightforward process of substitution once you have the values of 'a', 'b', and 'c'. In our case, we have:

• a = 1
• b = -2
• c = -15

Now, we substitute these values into the quadratic formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)}$

The next step is to simplify the expression. First, we simplify the terms inside the square root and the other parts of the equation:

$x = \frac{2 \pm \sqrt{4 + 60}}{2}$

$x = \frac{2 \pm \sqrt{64}}{2}$

Now, we can evaluate the square root of 64, which is 8:

$x = \frac{2 \pm 8}{2}$

At this point, we have two possible solutions, one with the '+' sign and one with the '-' sign. We will separate these into two distinct equations to find the two solutions for 'x'. This application of the quadratic formula demonstrates its power in providing a systematic approach to solving quadratic equations. The key is to carefully substitute the values and simplify the expression step by step.

Step 4: Simplify and Find the Two Solutions

Having applied the quadratic formula and simplified the expression to $x = \frac{2 \pm 8}{2}$, we now need to separate the '±' into two distinct equations to find the two possible solutions for 'x'. This is a crucial step because quadratic equations, by their nature, often have two solutions (although they can also have one repeated solution or two complex solutions). We will address each case separately:

Case 1: Using the '+' sign

$x = \frac{2 + 8}{2}$

We first add the numbers in the numerator:

$x = \frac{10}{2}$

Then, we divide to find the solution:

$x = 5$

So, one solution is x = 5.

Case 2: Using the '-' sign

$x = \frac{2 - 8}{2}$

We subtract the numbers in the numerator:

$x = \frac{-6}{2}$

Then, we divide to find the solution:

$x = -3$

So, the other solution is x = -3.

Therefore, the two solutions for the quadratic equation $x^2 - 2x = 15$ are x = 5 and x = -3. These are the values of 'x' that satisfy the original equation. This step demonstrates the importance of carefully considering both the '+' and '-' signs in the quadratic formula to ensure that we find all possible solutions. By separating the equation into two cases, we systematically arrive at the two roots of the quadratic equation.

Step 5: Verify the Solutions

After finding the solutions to a quadratic equation, it's always a good practice to verify the solutions. This step ensures that the values we obtained are indeed correct and satisfy the original equation. Verification helps to catch any potential errors made during the solution process. To verify our solutions, we substitute each value of 'x' back into the original equation and check if the equation holds true. Our original equation is $x^2 - 2x = 15$, and our solutions are $x = 5$ and $x = -3$.

Verification for x = 5

Substitute x = 5 into the equation:

$(5)^2 - 2(5) = 15$

Simplify the equation:

$25 - 10 = 15$

$15 = 15$

Since the equation holds true, x = 5 is a valid solution.

Verification for x = -3

Substitute x = -3 into the equation:

$(-3)^2 - 2(-3) = 15$

Simplify the equation:

$9 + 6 = 15$

$15 = 15$

Since the equation holds true, x = -3 is also a valid solution.

Both solutions, x = 5 and x = -3, satisfy the original equation. This verification step provides us with confidence that our solutions are correct. It's a simple yet powerful method to ensure accuracy in problem-solving. By verifying our solutions, we complete the process of solving the quadratic equation and confirm that we have found the correct roots.

Conclusion

In this comprehensive guide, we have successfully solved the quadratic equation $x^2 - 2x = 15$ using the quadratic formula. We meticulously walked through each step, from rewriting the equation in standard form to verifying the solutions. By understanding the underlying concepts and following the steps carefully, you can confidently tackle a wide range of quadratic equations. Remember, the key to success lies in accurately identifying the coefficients, correctly applying the quadratic formula, and diligently verifying your solutions. The quadratic formula is a powerful tool in mathematics, and mastering its application opens doors to solving various problems in different fields. We found the solutions to be x = 5 and x = -3, which corresponds to option B. Therefore, the correct answer is:

B. x = 5, x = -3

Keep practicing, and you'll become proficient in solving quadratic equations using the quadratic formula.