Solve For $x$ In The Equation $x^2 + 20x + 100 = 36$.A. $x = -16$ Or $x = -4$B. $x = -10$C. $x = -8$D. $x = 4$ Or $x = 16$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $x^2 + 20x + 100 = 36$, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, quadratic formula, and graphing.

The Given Equation

The given equation is $x^2 + 20x + 100 = 36$. To solve for $x$, we need to isolate the variable on one side of the equation. We can start by subtracting 36 from both sides of the equation, which gives us:

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

Simplifying the equation, we get:

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

Factoring the Equation

One of the methods to solve quadratic equations is by factoring. Factoring involves expressing the quadratic expression as a product of two binomials. In this case, we can factor the equation as:

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

Solving for $x$

To find the solutions, we need to set each factor equal to zero and solve for $x$. Setting the first factor equal to zero, we get:

$x + 16 = 0$

Subtracting 16 from both sides, we get:

$x = -16$

Setting the second factor equal to zero, we get:

$x + 4 = 0$

Subtracting 4 from both sides, we get:

$x = -4$

Conclusion

In conclusion, we have solved the quadratic equation $x^2 + 20x + 100 = 36$ using the factoring method. The solutions to the equation are $x = -16$ and $x = -4$. These solutions satisfy the original equation, and we can verify this by plugging them back into the equation.

Alternative Methods

While factoring is a powerful method for solving quadratic equations, it may not always be possible to factor the equation. In such cases, we can use alternative methods, such as the quadratic formula or graphing. The quadratic formula is a general method for solving quadratic equations, and it is given by:

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

In this case, we can plug in the values of $a$, $b$, and $c$ into the quadratic formula to find the solutions.

Quadratic Formula

Using the quadratic formula, we can find the solutions to the equation $x^2 + 20x + 100 = 36$. Plugging in the values of $a$, $b$, and $c$, we get:

$x = \frac{-20 \pm \sqrt{20^2 - 4(1)(64)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{-20 \pm \sqrt{400 - 256}}{2}$

$x = \frac{-20 \pm \sqrt{144}}{2}$

$x = \frac{-20 \pm 12}{2}$

Solving for $x$, we get:

$x = \frac{-20 + 12}{2}$

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

$x = -4$

$x = \frac{-20 - 12}{2}$

$x = \frac{-32}{2}$

$x = -16$

Graphing

Another method for solving quadratic equations is by graphing. Graphing involves plotting the quadratic function on a coordinate plane and finding the points where the function intersects the x-axis. These points represent the solutions to the equation.

Graphing the Equation

To graph the equation $x^2 + 20x + 100 = 36$, we can start by plotting the quadratic function on a coordinate plane. We can use a graphing calculator or software to plot the function.

Finding the Solutions

Once we have plotted the function, we can find the solutions by identifying the points where the function intersects the x-axis. These points represent the solutions to the equation.

Conclusion

In conclusion, we have solved the quadratic equation $x^2 + 20x + 100 = 36$ using three different methods: factoring, quadratic formula, and graphing. The solutions to the equation are $x = -16$ and $x = -4$. These solutions satisfy the original equation, and we can verify this by plugging them back into the equation.

Final Answer

The final answer is:

A. $x = -16$ or $x = -4$

$$$$

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a general method for solving quadratic equations. It is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula is a general method for solving quadratic equations, while factoring is a specific method that involves expressing the quadratic expression as a product of two binomials.

Q: Can I use graphing to solve quadratic equations?

A: Yes, you can use graphing to solve quadratic equations. Graphing involves plotting the quadratic function on a coordinate plane and finding the points where the function intersects the x-axis. These points represent the solutions to the equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or software. Plot the function on a coordinate plane and identify the points where the function intersects the x-axis.

Q: What are the advantages and disadvantages of each method?

A: Here are the advantages and disadvantages of each method:

• Factoring:
  • Advantages: Easy to use, simple to understand.
  • Disadvantages: May not always be possible to factor the equation.
• Quadratic Formula:
  • Advantages: General method, can be used for any quadratic equation.
  • Disadvantages: May be difficult to simplify the expression.
• Graphing:
  • Advantages: Visual representation, can be used to identify multiple solutions.
  • Disadvantages: May be difficult to plot the function, requires graphing calculator or software.

Q: Can I use a combination of methods to solve a quadratic equation?

A: Yes, you can use a combination of methods to solve a quadratic equation. For example, you can use factoring to simplify the equation and then use the quadratic formula to find the solutions.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Here are some common mistakes to avoid when solving quadratic equations:

• Not checking the solutions: Make sure to plug the solutions back into the original equation to verify that they are correct.
• Not simplifying the expression: Make sure to simplify the expression before solving for $x$.
• Not using the correct method: Choose the correct method for the specific equation and your personal preference.

Conclusion

In conclusion, solving quadratic equations requires a combination of mathematical skills and problem-solving strategies. By understanding the different methods and techniques, you can choose the best approach for each equation and solve them with confidence.