Completing The Square Common Core Algebra 1 Homework

# Completing the Square: A Simple and Effective Method for Common Core Algebra 1 Homework If you are struggling with quadratic functions and their algebra, you are not alone. Many students find this topic challenging and confusing. But don’t worry, there is a simple and effective method that can help you solve any quadratic equation and find the vertex of any parabola. It is called **completing the square**. ## What is Completing the Square? Completing the square is a technique that involves adding a constant term to both sides of a quadratic equation to make it into a perfect square trinomial. A perfect square trinomial is a polynomial that can be factored into the square of a binomial. For example, `x^2 + 6x + 9` is a perfect square trinomial because it can be factored into `(x + 3)^2`. Why do we want to make a quadratic equation into a perfect square trinomial? Because it makes it easier to solve for x and to find the vertex of the parabola. Let’s see how. ## How to Complete the Square To complete the square, we need to follow these steps: 1. Make sure the coefficient of `x^2` is 1. If not, divide both sides of the equation by the coefficient. 2. Move the constant term to the right side of the equation. 3. Find half of the coefficient of x and square it. This is the term we need to add to both sides of the equation to complete the square. 4. Factor the left side of the equation into a binomial squared. 5. Simplify the right side of the equation by combining like terms. 6. Take the square root of both sides of the equation and solve for x. Let’s see an example: Solve `x^2 + 8x – 20 = 0` by completing the square. Step 1: The coefficient of `x^2` is already 1, so we don’t need to do anything. Step 2: Move the constant term to the right side of the equation: `x^2 + 8x = 20` Step 3: Find half of the coefficient of x and square it: `(8/2)^2 = 4^2 = 16` This is the term we need to add to both sides of the equation: `x^2 + 8x + 16 = 20 + 16` Step 4: Factor the left side of the equation into a binomial squared: `(x + 4)^2 = 36` Step 5: Simplify the right side of the equation by combining like terms: `(x + 4)^2 = 36` Step 6: Take the square root of both sides of the equation and solve for x: `x + 4 = ±√36` `x = -4 ±√36` `x = -4 ±6` `x = -10` or `x = 2` These are the solutions of the quadratic equation. ## How to Find the Vertex of a Parabola Completing the square can also help us find the vertex of a parabola. The vertex is the point where the parabola changes direction and has either a maximum or a minimum value. To find it, we need to use this formula: `y = a(x – h)^2 + k` where `(h, k)` is the vertex and `a` is a constant that determines how wide or narrow the parabola is. To use this formula, we need to complete the square on any quadratic function and write it in this form. Let’s see an example: Find the vertex of `y = x^2 – 6x + 5`. Step 1: Complete the square on `y = x^2 – 6x + 5`. Move the constant term to the right side: `y – 5 = x^2 – 6x` Find half of the coefficient of x and square it: `(-6/2)^2 = (-3)^2 = 9` Add this term to both sides: `y – 5 + 9 = x^2 – 6x + 9` Factor the right side into a binomial squared: `y + 4 = (x – 3)^2` Step 2: Write `y + 4 = (x – 3)^2` in vertex form by subtracting 4 from both sides: `y = (x – 3)^2 – 4` Now we can see that `(h, k) = (3, -4)` and `a = 1`. Therefore, **the vertex of this parabola is (3, -4)**. ## Why Completing The Square Is Useful for Common Core Algebra 1 Homework Completing

## How to Graph a Parabola Using Completing the Square Another benefit of completing the square is that it can help us graph a parabola easily. To graph a parabola, we need to know its vertex, its axis of symmetry, and its y-intercept. We can find all these information by completing the square and using the vertex form of a quadratic function. Let’s see an example: Graph `y = -2x^2 + 12x – 13` by completing the square. Step 1: Complete the square on `y = -2x^2 + 12x – 13`. Divide both sides by -2 to make the coefficient of `x^2` equal to 1: `-y/2 = x^2 – 6x + 13/2` Move the constant term to the left side: `-y/2 – 13/2 = x^2 – 6x` Find half of the coefficient of x and square it: `(-6/2)^2 = (-3)^2 = 9` Add this term to both sides: `-y/2 – 13/2 + 9 = x^2 – 6x + 9` Factor the right side into a binomial squared: `-y/2 + 4 = (x – 3)^2` Step 2: Write `-y/2 + 4 = (x – 3)^2` in vertex form by multiplying both sides by -2 and subtracting 8 from both sides: `y = -2(x – 3)^2 + 8` Now we can see that `(h, k) = (3, 8)` and `a = -2`. Therefore, **the vertex of this parabola is (3, 8)** and **the parabola opens downward** because `a < 0`. Step 3: Find the axis of symmetry and the y-intercept of the parabola. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The equation of the axis of symmetry is `x = h`, where `h` is the x-coordinate of the vertex. In this case, **the axis of symmetry is `x = 3`**. The y-intercept is the point where the parabola crosses the y-axis. To find it, we need to plug in `x = 0` into the original equation and solve for y. In this case, `y = -2(0)^2 + 12(0) – 13` `y = -13` Therefore, **the y-intercept is (0, -13)**. Step 4: Plot the vertex, the axis of symmetry, and the y-intercept on a coordinate plane and sketch the parabola. ![graph](https://i.imgur.com/7fOYwZl.png) This is how we can graph a parabola using completing the square. ## How to Apply Completing The Square to Real-World Problems Completing the square can also help us solve real-world problems that involve quadratic functions. For example, we can use completing the square to find the maximum or minimum value of a function, such as profit, area, height, etc. We can also use completing the square to model projectile motion, such as throwing a ball or shooting a cannon. Let's see an example: A farmer wants to fence a rectangular field with one side along a river. He has 1200 feet of fencing material and wants to maximize the area of his field. What are the dimensions of his field? Step 1: Define variables and write an equation. Let x be the length of each side perpendicular to the river and y be ## How to Check Your Work Using Completing the Square Completing the square can also help us check our work when solving quadratic equations by other methods, such as factoring or using the quadratic formula. To check our work, we need to complete the square on the original equation and compare it with our solutions. If our solutions make the equation into a perfect square trinomial, then we know we are correct. Let's see an example: Solve `x^2 – 10x + 16 = 0` by factoring and check your work by completing the square. Step 1: Solve by factoring. To factor a quadratic equation, we need to find two numbers that multiply to the constant term and add to the coefficient of x. In this case, we need to find two numbers that multiply to 16 and add to -10. These numbers are -8 and -2. Therefore, we can factor the equation as follows: `x^2 – 10x + 16 = 0` `(x – 8)(x – 2) = 0` To find the solutions, we need to set each factor equal to zero and solve for x: `x – 8 = 0` `x = 8` `x – 2 = 0` `x = 2` These are the solutions of the quadratic equation. Step 2: Check your work by completing the square. To check our work, we need to complete the square on the original equation and see if it matches our solutions. To complete the square, we need to follow these steps: 1. Make sure the coefficient of `x^2` is 1. If not, divide both sides of the equation by the coefficient. 2. Move the constant term to the right side of the equation. 3. Find half of the coefficient of x and square it. This is the term we need to add to both sides of the equation to complete the square. 4. Factor the left side of the equation into a binomial squared. 5. Simplify the right side of the equation by combining like terms. 6. Take the square root of both sides of the equation and solve for x. Let's see how it works: Step 1: The coefficient of `x^2` is already 1, so we don't need to do anything. Step 2: Move the constant term to the right side of the equation: `x^2 – 10x = -16` Step 3: Find half of the coefficient of x and square it: `(-10/2)^2 = (-5)^2 = 25` This is the term we need to add to both sides of the equation: `x^2 – 10x + 25 = -16 + 25` Step 4: Factor the left side of the equation into a binomial squared: `(x – 5)^2 = 9` Step 5: Simplify ## How to Practice Completing the Square for Common Core Algebra 1 Homework Completing the square is an important skill that can help you solve quadratic equations, find the vertex of a parabola, graph a parabola, and apply quadratic functions to real-world problems. To master this skill, you need to practice it regularly and check your work for accuracy. Here are some tips on how to practice completing the square for common core algebra 1 homework: – Review the steps and examples of completing the square before you start your homework. Make sure you understand the logic and the purpose of each step. – Follow the steps carefully and write down each step clearly. Show your work and label your calculations. – Check your work by plugging in your solutions into the original equation and see if they make it true. If not, go back and find your mistake. – Try different methods of solving quadratic equations and compare them with completing the square. See which method works best for different types of equations and situations. – Challenge yourself with more difficult or complex equations that require completing the square. For example, try equations that have fractions, decimals, negative coefficients, or irrational or complex solutions. – Use online resources to find more practice problems and solutions. For example, you can use Khan Academy, Flipped Math, or eMATHinstruction to find more videos, worksheets, and homework on completing the square. Completing the square is a useful and powerful technique that can help you succeed in common core algebra 1. With enough practice and confidence, you can master this skill and apply it to various problems and situations. # Conclusion In this article, we have learned what completing the square is, how to complete the square, how to find the vertex of a parabola, how to graph a parabola, how to apply completing the square to real-world problems, how to compare completing the square with other methods, and how to practice completing the square for common core algebra 1 homework. Completing the square is a versatile and effective method that can help us solve and understand quadratic functions and their algebra. By following the steps and examples in this article, we can improve our skills and confidence in completing the square and use it to achieve our academic goals.

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