Solving Equations In The Set Of Real Numbers 2x + 3 = 11 And X² - 5x + 6 = 0

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Hey guys! Today, we're diving into the exciting world of solving equations within the set of real numbers (R). We'll be tackling two equations in particular: a linear equation and a quadratic equation. Don't worry if these terms sound intimidating – we'll break everything down step-by-step, making it super easy to understand. So, grab your pencils and notebooks, and let's get started!

1) Solving the Linear Equation: 2x + 3 = 11

Let's kick things off with our first equation, a classic linear equation: 2x + 3 = 11. Linear equations are those where the highest power of the variable (in this case, 'x') is 1. They represent a straight line when graphed, hence the name 'linear'.

Isolating the Variable: The Key to Success

The main goal when solving any equation is to isolate the variable – that is, to get the variable all by itself on one side of the equation. Think of it like separating a stubborn toddler from their favorite toy! We want to get 'x' alone, so we need to undo any operations that are being done to it.

In our equation, 'x' is being multiplied by 2 and then having 3 added to the result. To isolate 'x', we need to reverse these operations, following the order of operations in reverse (PEMDAS/BODMAS – Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). So, we'll first get rid of the addition, and then the multiplication.

Step 1: Subtracting 3 from Both Sides

The first step is to subtract 3 from both sides of the equation. Why both sides? Because we need to maintain the equality. Think of an equation like a balanced scale. If you add or subtract something from one side, you need to do the same on the other side to keep it balanced. So, we have:

2x + 3 - 3 = 11 - 3

This simplifies to:

2x = 8

We've successfully gotten rid of the '+ 3' on the left side, and 'x' is one step closer to being isolated!

Step 2: Dividing Both Sides by 2

Now, 'x' is being multiplied by 2. To undo this multiplication, we need to divide both sides of the equation by 2. Again, we're doing the same operation to both sides to maintain balance:

2x / 2 = 8 / 2

This simplifies to:

x = 4

Ta-da! We've solved the equation! The solution for 'x' in the equation 2x + 3 = 11 is x = 4. This means that if we substitute '4' for 'x' in the original equation, it will hold true: 2(4) + 3 = 8 + 3 = 11. It checks out!

2) Solving the Quadratic Equation: x² - 5x + 6 = 0

Next up, we have a quadratic equation: x² - 5x + 6 = 0. Quadratic equations are those where the highest power of the variable is 2. They have a slightly different flavor than linear equations, and solving them requires a different set of techniques.

Understanding Quadratic Equations

A quadratic equation has the general form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants. In our case, a = 1, b = -5, and c = 6. The solutions to a quadratic equation are also called its roots or zeros. These are the values of 'x' that make the equation true.

Unlike linear equations, quadratic equations can have up to two distinct solutions. This is because the graph of a quadratic equation is a parabola, which can intersect the x-axis at two points, one point, or no points at all.

Methods for Solving Quadratic Equations

There are a few common methods for solving quadratic equations:

  • Factoring: This involves rewriting the quadratic expression as a product of two linear expressions.
  • Quadratic Formula: This is a general formula that gives the solutions for any quadratic equation.
  • Completing the Square: This is a method that transforms the quadratic equation into a perfect square trinomial.

For this particular equation, factoring will be the easiest and fastest method. Let's dive in!

Factoring the Quadratic Expression

The goal of factoring is to rewrite the expression x² - 5x + 6 as a product of two binomials (expressions with two terms). We're looking for two numbers that:

  • Multiply to give the constant term (c = 6).
  • Add up to give the coefficient of the x term (b = -5).

Let's think about the factors of 6: 1 and 6, 2 and 3. Since we need the numbers to add up to -5, we need to consider negative factors. The pair that works is -2 and -3 because (-2) * (-3) = 6 and (-2) + (-3) = -5.

So, we can rewrite the quadratic expression as:

(x - 2)(x - 3) = 0

Applying the Zero Product Property

Now we come to a crucial concept: the Zero Product Property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In other words, if A * B = 0, then either A = 0 or B = 0 (or both).

Applying this to our factored equation, (x - 2)(x - 3) = 0, we know that either (x - 2) = 0 or (x - 3) = 0.

Solving for x

Now we have two simple linear equations to solve:

  • x - 2 = 0
  • x - 3 = 0

Adding 2 to both sides of the first equation gives us:

x = 2

Adding 3 to both sides of the second equation gives us:

x = 3

So, the solutions for 'x' in the quadratic equation x² - 5x + 6 = 0 are x = 2 and x = 3. This means that if we substitute either 2 or 3 for 'x' in the original equation, it will hold true. You can try it out and see!

Final Answer and Choosing the Correct Alternative

Alright, we've successfully solved both equations! Let's recap our findings:

  • For the linear equation 2x + 3 = 11, the solution is x = 4.
  • For the quadratic equation x² - 5x + 6 = 0, the solutions are x = 2 and x = 3.

Now, let's take a look at the multiple-choice options provided:

  • a) x = 4 and x = 2
  • b) x = 3 and x = 2
  • c) x = 5 and x = 1
  • d) x = 6 and x = 0

Comparing our solutions to the options, we can see that the correct alternative is b) x = 3 and x = 2. This option correctly lists the solutions for the quadratic equation. While x = 4 is a solution we found, it corresponds to the linear equation, not the quadratic one.

Conclusion: Mastering Equation Solving

Great job, guys! We've tackled both a linear and a quadratic equation, and we've seen how different techniques are used to solve them. Remember, the key to solving equations is to isolate the variable by performing inverse operations on both sides of the equation. With practice, you'll become a pro at solving all sorts of equations!

Solving equations is a fundamental skill in mathematics, and it's used in countless applications, from physics and engineering to economics and computer science. So, keep practicing, and don't be afraid to ask questions. You've got this!