Explanation And Solution For "What Is The Positive Solution Of X<Sup>2</Sup> + 36 = 5X"

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Introduction

As a professional teacher, one of the most common questions I receive from students is how to solve quadratic equations. One particular problem that students often struggle with is the equation x² + 36 = 5x. In this article, I will explain the steps needed to find the positive solution to this equation in relaxed English language.

Understanding Quadratic Equations

Before we dive into the solution to this particular equation, it's important to have a basic understanding of quadratic equations. A quadratic equation is an equation of the second degree, meaning that it contains one or more terms that are squared. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and x is the variable.

Step 1: Rewrite the Equation in Standard Form

The first step in solving any quadratic equation is to rewrite it in standard form, which is ax² + bx + c = 0. In the case of x² + 36 = 5x, we need to move all the terms to one side of the equation to get it in standard form. Subtracting 5x from both sides, we get x² - 5x + 36 = 0.

Step 2: Identify the Values of a, b, and c

The next step is to identify the values of a, b, and c in the standard form equation. In our equation, a = 1, b = -5, and c = 36.

Step 3: Use the Quadratic Formula

Now that we have the values of a, b, and c, we can use the quadratic formula to solve for x. The quadratic formula is x = (-b ± sqrt(b² - 4ac)) / 2a. Plugging in the values from our equation, we get x = (5 ± sqrt((-5)² - 4(1)(36))) / 2(1).

Step 4: Simplify the Quadratic Formula

To simplify the quadratic formula, we need to evaluate the expression inside the square root. (-5)² - 4(1)(36) simplifies to 25 - 144, which equals -119. Since we can't take the square root of a negative number, we know that there are no real solutions to the equation.

Step 5: Finding the Imaginary Solution

However, there are still solutions to the equation in the form of imaginary numbers. To find the imaginary solution, we need to simplify the expression inside the square root by factoring out -1. This gives us x = (5 ± i sqrt(119)) / 2. The positive solution is (5 + i sqrt(119)) / 2.

Conclusion

In conclusion, the positive solution to x² + 36 = 5x is (5 + i sqrt(119)) / 2. While this equation may seem daunting at first, breaking it down into steps and using the quadratic formula can help to simplify the process. Remember to always check for both real and imaginary solutions when solving quadratic equations.