Solving $x^2 + 20x + 100 = 36$ A Comprehensive Guide

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In the realm of mathematics, solving equations is a fundamental skill. Quadratic equations, in particular, play a significant role in various fields, from physics to engineering. This article delves into the process of solving the quadratic equation $x^2 + 20x + 100 = 36$, providing a step-by-step guide and exploring the underlying concepts.

Understanding Quadratic Equations

Before we dive into the solution, it's crucial to understand what quadratic equations are. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. The solutions to a quadratic equation are also known as its roots or zeros.

Quadratic equations can have two, one, or no real solutions, depending on the discriminant ($b^2 - 4ac$). The discriminant provides valuable information about the nature of the roots:

• If $b^2 - 4ac > 0$, the equation has two distinct real roots.
• If $b^2 - 4ac = 0$, the equation has one real root (a repeated root).
• If $b^2 - 4ac < 0$, the equation has no real roots (two complex roots).

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, each with its advantages and disadvantages. The most common methods include:

1. Factoring: This method involves expressing the quadratic equation as a product of two linear factors. It is efficient for equations that can be easily factored.

2. Completing the Square: This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve.

3. Quadratic Formula: This formula provides a general solution for any quadratic equation, regardless of its factorability. The quadratic formula is given by:

   x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}

Solving $x^2 + 20x + 100 = 36$ Using Multiple Approaches

Now, let's tackle the equation $x^2 + 20x + 100 = 36$ using different methods to illustrate the versatility of problem-solving techniques in mathematics.

1. Solving by Factoring: A Detailed Approach

Factoring is a powerful technique for solving quadratic equations, particularly when the equation can be expressed as a product of two binomials. In this section, we'll demonstrate how to solve the equation $x^2 + 20x + 100 = 36$ by factoring, breaking down each step for clarity.

Step 1: Rearrange the Equation

The first crucial step in solving any quadratic equation is to set it equal to zero. This allows us to apply factoring or other methods effectively. To do this, we subtract 36 from both sides of the equation:

$x^2 + 20x + 100 - 36 = 36 - 36$

This simplifies to:

$x^2 + 20x + 64 = 0$

Now we have a standard quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = 20$, and $c = 64$.

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression $x^2 + 20x + 64$. We're looking for two numbers that multiply to $c$ (64) and add up to $b$ (20). Let's consider the factors of 64:

• 1 and 64
• 2 and 32
• 4 and 16
• 8 and 8

Among these pairs, 4 and 16 add up to 20, which is our desired sum. Therefore, we can rewrite the quadratic expression as:

$x^2 + 4x + 16x + 64$

Now, we factor by grouping:

$x(x + 4) + 16(x + 4)$

Notice that $(x + 4)$ is a common factor, so we can factor it out:

$(x + 4)(x + 16)$

Thus, our factored equation is:

$(x + 4)(x + 16) = 0$

Step 3: Set Each Factor Equal to Zero

The principle behind this step is the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero:

$x + 4 = 0$ or $x + 16 = 0$

Step 4: Solve for $x$

Now, we solve each linear equation for $x$:

For $x + 4 = 0$, subtract 4 from both sides:

$x = -4$

For $x + 16 = 0$, subtract 16 from both sides:

$x = -16$

Step 5: Verify the Solutions

It's always a good practice to verify our solutions by substituting them back into the original equation to ensure they satisfy it.

Let's check $x = -4$:

$(-4)^2 + 20(-4) + 100 = 16 - 80 + 100 = 36$

This is true, so $x = -4$ is a valid solution.

Now let's check $x = -16$:

$(-16)^2 + 20(-16) + 100 = 256 - 320 + 100 = 36$

This is also true, so $x = -16$ is a valid solution.

Conclusion: Solutions by Factoring

By factoring the quadratic equation $x^2 + 20x + 100 = 36$, we found two solutions: $x = -4$ and $x = -16$. This method demonstrates how factoring can efficiently solve quadratic equations when the equation can be factored easily.

2. Solving by Completing the Square: A Step-by-Step Guide

Completing the square is a versatile method for solving quadratic equations. It involves transforming the equation into a perfect square trinomial. This section provides a detailed walkthrough of solving $x^2 + 20x + 100 = 36$ using this technique.

Step 1: Rearrange the Equation

As with the factoring method, we begin by setting the equation equal to zero:

$x^2 + 20x + 100 - 36 = 0$

Simplifying, we get:

$x^2 + 20x + 64 = 0$

Step 2: Move the Constant Term to the Right Side

Next, we move the constant term (64) to the right side of the equation by subtracting it from both sides:

$x^2 + 20x = -64$

Step 3: Complete the Square

To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. The value we need to add is (rac{b}{2})^2, where $b$ is the coefficient of the $x$ term. In this case, $b = 20$, so we calculate:

(rac{20}{2})^2 = (10)^2 = 100

Now, we add 100 to both sides of the equation:

$x^2 + 20x + 100 = -64 + 100$

Simplifying, we get:

$x^2 + 20x + 100 = 36$

Step 4: Factor the Left Side

The left side of the equation is now a perfect square trinomial, which can be factored as $(x + 10)^2$:

$(x + 10)^2 = 36$

Step 5: Take the Square Root of Both Sides

To eliminate the square on the left side, we take the square root of both sides of the equation. Remember to consider both positive and negative square roots:

$ ext{√}((x + 10)^2) = ext{±} ext{√}(36)$

This gives us:

$x + 10 = ext{±} 6$

Step 6: Solve for $x$

Now, we solve for $x$ by considering both cases:

Case 1: $x + 10 = 6$

Subtract 10 from both sides:

$x = 6 - 10$

$x = -4$

Case 2: $x + 10 = -6$

Subtract 10 from both sides:

$x = -6 - 10$

$x = -16$

Step 7: Verify the Solutions

We verify the solutions by substituting them back into the original equation, just as we did with the factoring method.

For $x = -4$:

$(-4)^2 + 20(-4) + 100 = 16 - 80 + 100 = 36$

For $x = -16$:

$(-16)^2 + 20(-16) + 100 = 256 - 320 + 100 = 36$

Both solutions are valid.

Conclusion: Solutions by Completing the Square

Using the completing the square method, we have found the solutions to the quadratic equation $x^2 + 20x + 100 = 36$ to be $x = -4$ and $x = -16$. This method is particularly useful for equations that are not easily factored and provides a systematic approach to finding solutions.

3. Solving with the Quadratic Formula: A Universal Approach

The quadratic formula is a powerful and versatile tool for solving any quadratic equation in the form $ax^2 + bx + c = 0$. It provides a direct method to find the solutions, regardless of whether the equation can be factored or easily transformed. In this section, we'll use the quadratic formula to solve the equation $x^2 + 20x + 100 = 36$.

Step 1: Rearrange the Equation

As with the other methods, we first set the equation equal to zero:

$x^2 + 20x + 100 - 36 = 0$

Simplifying, we get:

$x^2 + 20x + 64 = 0$

Step 2: Identify $a$, $b$, and $c$

In the general form of a quadratic equation $ax^2 + bx + c = 0$, we identify the coefficients $a$, $b$, and $c$. In our equation, $x^2 + 20x + 64 = 0$, we have:

• $a = 1$
• $b = 20$
• $c = 64$

Step 3: Apply the Quadratic Formula

The quadratic formula is given by:

x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}

Substitute the values of $a$, $b$, and $c$ into the formula:

x = rac{-20 ext{±} ext{√}(20^2 - 4(1)(64))}{2(1)}

Step 4: Simplify the Expression

Now, we simplify the expression step by step:

First, calculate the value inside the square root:

$20^2 - 4(1)(64) = 400 - 256 = 144$

So, the equation becomes:

x = rac{-20 ext{±} ext{√}(144)}{2}

Next, we find the square root of 144:

$ ext{√}(144) = 12$

Thus, the equation simplifies to:

x = rac{-20 ext{±} 12}{2}

Step 5: Solve for $x$

We now have two cases to consider:

Case 1: Using the positive square root:

x = rac{-20 + 12}{2}

x = rac{-8}{2}

$x = -4$

Case 2: Using the negative square root:

x = rac{-20 - 12}{2}

x = rac{-32}{2}

$x = -16$

Step 6: Verify the Solutions

As with the other methods, we verify the solutions by substituting them back into the original equation.

For $x = -4$:

$(-4)^2 + 20(-4) + 100 = 16 - 80 + 100 = 36$

For $x = -16$:

$(-16)^2 + 20(-16) + 100 = 256 - 320 + 100 = 36$

Both solutions are valid.

Conclusion: Solutions with the Quadratic Formula

Using the quadratic formula, we've determined the solutions to the equation $x^2 + 20x + 100 = 36$ to be $x = -4$ and $x = -16$. The quadratic formula is a reliable method for solving quadratic equations, especially when other methods are not straightforward.

Conclusion

In this article, we explored various methods for solving the quadratic equation $x^2 + 20x + 100 = 36$. We demonstrated how to solve the equation by factoring, completing the square, and using the quadratic formula. Each method provides a unique approach, highlighting the flexibility of mathematical problem-solving. The solutions to the equation are $x = -4$ and $x = -16$, which were verified using each method. Understanding these methods enhances one's ability to tackle a wide range of quadratic equations and reinforces fundamental algebraic concepts.

Whether you prefer factoring for its efficiency, completing the square for its systematic approach, or the quadratic formula for its generality, mastering these techniques is crucial for success in mathematics and related fields. By practicing and understanding these methods, you can confidently solve quadratic equations and apply these skills to more complex problems.