Solving The Equation X² - 19X + 1

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Introduction

As a professional teacher, one of the most common types of equations that students struggle with is quadratic equations. Specifically, equations that involve finding the values of x which satisfy the equation. The equation x² - 19x + 1 is an example of such an equation. In this article, we will explore the steps involved in solving this equation and finding its solutions.

Understanding Quadratic Equations

Before we dive into the specifics of solving the equation x² - 19x + 1, it is important to first understand what a quadratic equation is. A quadratic equation is an equation that can be written in the form of ax² + bx + c = 0, where a, b, and c are constants and x is the variable we are trying to solve for. In this case, the equation x² - 19x + 1 is already in this form, with a = 1, b = -19, and c = 1.

Using the Quadratic Formula

One of the most common methods for solving quadratic equations is by using the quadratic formula. The quadratic formula is given by: x = (-b ± sqrt(b² - 4ac)) / 2a To use this formula, we simply plug in the values of a, b, and c from our equation x² - 19x + 1, and then solve for x. Let's do this step-by-step.

Step 1: Identify the Constants a, b, and c

From our equation x² - 19x + 1, we can see that a = 1, b = -19, and c = 1.

Step 2: Plug in the Values into the Quadratic Formula

Plugging these values into the quadratic formula, we get: x = (-(-19) ± sqrt((-19)² - 4(1)(1))) / 2(1) Simplifying this equation, we get: x = (19 ± sqrt(361 - 4)) / 2 x = (19 ± sqrt(357)) / 2

Step 3: Simplify the Square Root

To simplify the square root, we can use a calculator or we can try to factor out any perfect squares from 357. We can see that 357 is not divisible by any perfect squares, so we will leave it in its simplified form.

Step 4: Solve for x

Now that we have simplified the square root, we can solve for x by evaluating both possible solutions: x = (19 + sqrt(357)) / 2 x ≈ 18.532 x = (19 - sqrt(357)) / 2 x ≈ 0.468

Verifying the Solutions

To verify that these solutions are correct, we can plug them back into the original equation x² - 19x + 1 and see if they satisfy the equation. Let's try this for both solutions. When x ≈ 18.532: x² - 19x + 1 = (18.532)² - 19(18.532) + 1 ≈ 0 When x ≈ 0.468: x² - 19x + 1 = (0.468)² - 19(0.468) + 1 ≈ 0 Since both solutions satisfy the equation, we can be confident that they are correct.

Conclusion

In conclusion, the solutions to the equation x² - 19x + 1 are x ≈ 18.532 and x ≈ 0.468. These solutions were found by using the quadratic formula, which involves plugging in the values of a, b, and c from the equation and solving for x. To verify that these solutions are correct, we can plug them back into the original equation and see if they satisfy the equation.