Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers

[Edelira Longinotti]

Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers

In this article, we will review how to find the zeros of a quadratic function by completing the square, a technique that can help us rewrite a quadratic expression in the form (x + a)^2 + b. We will also look at some examples and practice problems from the Common Core Algebra 1 curriculum.

What is completing the square?

Completing the square is a technique for rewriting quadratics in the form (x + a)^2 + b. For example, x^2 + 2x + 3 can be rewritten as (x + 1)^2 + 2. The two expressions are totally equivalent, but the second one is nicer to work with in some situations.

Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers

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To complete the square, we need to follow these steps:

• If the coefficient of the x^2 term is not 1, divide the polynomial by that coefficient.

• Move the constant term to the right side of the equation.

• Add the square of half of the coefficient of the x term to both sides of the equation.

• Factor the left side of the equation as a perfect square.

• Solve for x by taking the square root of both sides and isolating x.

Why do we complete the square?

One of the main purposes of completing the square is to find the zeros of a quadratic function, which are the values of x that make the function equal to zero. Finding zeros is useful for graphing, modeling, and solving real-world problems involving quadratics.

Another purpose of completing the square is to get an equation in vertex form, which is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Vertex form makes it easy to identify the vertex, axis of symmetry, and direction of opening of a quadratic graph.

Examples and practice problems

Let's look at some examples and practice problems from the Common Core Algebra 1 curriculum that involve completing the square. We will use facts from [this video] and [this article] as references.

Example 1

We're given a quadratic and asked to complete the square.

x^2 + 10x + 24 = 0 

We begin by moving the constant term to the right side of the equation.

x^2 + 10x = -24 

We complete the square by taking half of the coefficient of our x term, squaring it, and adding it to both sides of the equation. Since the coefficient of our x term is 10, half of it would be 5, and squaring it gives us 25.

x^2 + 10x + 25 = -24 + 25 

We can now rewrite the left side of the equation as a squared term.

(x + 5)^2 = 1 

Take the square root of both sides.

x + 5 = 1 

Isolate x to find the solution(s).

x = -5  1 

The solutions are: x = -4 and x = -6. These are also the zeros of the quadratic function.

Example 2

We're given a quadratic and asked to complete the square.

4x^2 + 20x + 25 = 0 

First, divide the polynomial by 4 (the coefficient of the x^2 term).

x^2 + 5x + 6.25 = 0 

Note that the left side of the equation is already a perfect square trinomial. The coefficient of our x term is 5, half of it is 2.5, and squaring it gives us 6.25, our constant term. Thus, we can rewrite the left side of the equation as a squared term.

(x + 2.5)^2 = 0 

Take the square root of both sides.

x + 2.5 = 0 

Isolate x to find the solution. The solution is: x = -2.5. This is also the zero of the quadratic function.

Practice Problem 1

Complete the square to rewrite this expression in the form (x + a)^2 + b.

x^2 - 2x + 17 

To complete the square, we need to add the square of half of the coefficient of the x term to both sides of the equation. Since the coefficient of our x term is -2, half of it would be -1, and squaring it gives us 1.

x^2 - 2x + 1 = 17 + 1 

We can now rewrite the left side of the equation as a squared term.

(x - 1)^2 = 18 

This is the desired form. We can see that a = -1 and b = 18.

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