Finding the Zeros of the Quadratic Function

Oct 13, 2025 by miodeya.com 49 views

Introduction

In algebra, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $a$ is not equal to zero. The zeros of a quadratic function are the values of $x$ that make the function equal to zero. In this article, we will learn how to find the zeros of a quadratic function by completing the square.

What are the Zeros of a Quadratic Function?

The zeros of a quadratic function are the values of $x$ that make the function equal to zero. In other words, if we substitute the zero of a quadratic function into the function, the result will be zero. For example, if we have a quadratic function $f(x) = x^2 - 4x + 4$ , the zeros of this function are $x = 2$ and $x = 2$ . This is because when we substitute $x = 2$ into the function, we get $f(2) = (2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$ , and when we substitute $x = 2$ into the function, we get $f(2) = (2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$ .

Completing the Square

Completing the square is a technique used to rewrite a quadratic function in a form that makes it easier to find the zeros. The basic idea behind completing the square is to rewrite the quadratic function in the form $f(x) = a(x - h)^2 + k$ , where $h$ and $k$ are constants. This form is called the vertex form of a quadratic function.

To complete the square, we start by writing the quadratic function in the form $f(x) = ax^2 + bx + c$ . We then group the terms with the variable $x$ together, and move the constant term to the right-hand side of the equation. We then add and subtract a constant term to the left-hand side of the equation, so that the left-hand side of the equation is a perfect square trinomial.

Step 1: Write the Quadratic Function in the Form $f(x) = ax^2 + bx + c$

The given quadratic function is $f(x) = 6x^2 - 24x + 1$ . We can write this function in the form $f(x) = ax^2 + bx + c$ by simply writing it as $f(x) = 6x^2 - 24x + 1$ .

Step 2: Group the Terms with the Variable $x$ Together

We can group the terms with the variable $x$ together by writing the function as $f(x) = 6x^2 - 24x + 1 = 6(x^2 - 4x) + 1$ .

Step 3: Move the Constant Term to the Right-Hand Side of the Equation

We can move the constant term to the right-hand side of the equation by writing the function as $f(x) = 6(x^2 - 4x) + 1 = 6(x^2 - 4x) + 1 - 1 = 6(x^2 - 4x) - 1$ .

Step 4: Add and Subtract a Constant Term to the Left-Hand Side of the Equation

We can add and subtract a constant term to the left-hand side of the equation by writing the function as $f(x) = 6(x^2 - 4x) - 1 = 6(x^2 - 4x + 4) - 1 - 24 = 6(x^2 - 4x + 4) - 25$ .

Step 5: Factor the Perfect Square Trinomial

We can factor the perfect square trinomial by writing the function as $f(x) = 6(x^2 - 4x + 4) - 25 = 6(x - 2)^2 - 25$ .

Step 6: Write the Function in the Vertex Form

We can write the function in the vertex form by writing it as $f(x) = 6(x - 2)^2 - 25 = 6(x - 2)^2 - 25 = 6(x - 2)^2 - 25 + 25 = 6(x - 2)^2$ .

Finding the Zeros of the Quadratic Function

Now that we have written the quadratic function in the vertex form, we can find the zeros of the function by setting the function equal to zero and solving for $x$ . We can set the function equal to zero by writing the equation $6(x - 2)^2 = 0$ . We can solve for $x$ by taking the square root of both sides of the equation, which gives us $x - 2 = 0$ . We can then add 2 to both sides of the equation, which gives us $x = 2$ .

Conclusion

In this article, we learned how to find the zeros of a quadratic function by completing the square. We started by writing the quadratic function in the form $f(x) = ax^2 + bx + c$ . We then grouped the terms with the variable $x$ together, moved the constant term to the right-hand side of the equation, added and subtracted a constant term to the left-hand side of the equation, factored the perfect square trinomial, and wrote the function in the vertex form. We then found the zeros of the function by setting the function equal to zero and solving for $x$ . We can use this technique to find the zeros of any quadratic function.

Example

Let's consider the quadratic function $f(x) = x^2 - 6x + 8$ . We can find the zeros of this function by completing the square. We start by writing the function in the form $f(x) = ax^2 + bx + c$ . We then group the terms with the variable $x$ together, move the constant term to the right-hand side of the equation, add and subtract a constant term to the left-hand side of the equation, factor the perfect square trinomial, and write the function in the vertex form. We then find the zeros of the function by setting the function equal to zero and solving for $x$ .

Step 1: Write the Quadratic Function in the Form $f(x) = ax^2 + bx + c$

The given quadratic function is $f(x) = x^2 - 6x + 8$ . We can write this function in the form $f(x) = ax^2 + bx + c$ by simply writing it as $f(x) = x^2 - 6x + 8$ .

Step 2: Group the Terms with the Variable $x$ Together

We can group the terms with the variable $x$ together by writing the function as $f(x) = x^2 - 6x + 8 = (x^2 - 6x) + 8$ .

Step 3: Move the Constant Term to the Right-Hand Side of the Equation

We can move the constant term to the right-hand side of the equation by writing the function as $f(x) = (x^2 - 6x) + 8 = (x^2 - 6x) + 8 - 8 = (x^2 - 6x) - 8$ .

Step 4: Add and Subtract a Constant Term to the Left-Hand Side of the Equation

We can add and subtract a constant term to the left-hand side of the equation by writing the function as $f(x) = (x^2 - 6x) - 8 = (x^2 - 6x + 9) - 8 - 9 = (x^2 - 6x + 9) - 17$ .

Step 5: Factor the Perfect Square Trinomial

We can factor the perfect square trinomial by writing the function as $f(x) = (x^2 - 6x + 9) - 17 = (x - 3)^2 - 17$ .

Step 6: Write the Function in the Vertex Form

We can write the function in the vertex form by writing it as $f(x) = (x - 3)^2 - 17 = (x - 3)^2 - 17 = (x - 3)^2 - 17 + 17 = (x - 3)^2$ .

Finding the Zeros of the Quadratic Function

Now that we have written the quadratic function in the vertex form, we can find the zeros of the function by setting the function equal to zero and solving for $x$ . We can set the function equal to zero by writing the equation $(x - 3)^2 = 0$ . We can solve for $x$ by taking the square root of both sides of the equation, which gives us $x - 3 = 0$ . We can then add 3 to both sides of the equation, which gives us $x = 3$ .

Conclusion

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic function in a form that makes it easier to find the zeros. The basic idea behind completing the square is to rewrite the quadratic function in the form $f(x) = a(x - h)^2 + k$ , where $h$ and $k$ are constants.

Q: How do I complete the square?

A: To complete the square, you start by writing the quadratic function in the form $f(x) = ax^2 + bx + c$ . You then group the terms with the variable $x$ together, move the constant term to the right-hand side of the equation, add and subtract a constant term to the left-hand side of the equation, factor the perfect square trinomial, and write the function in the vertex form.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$ , where $h$ and $k$ are constants. This form is also known as the standard form of a quadratic function.

Q: How do I find the zeros of a quadratic function by completing the square?

A: To find the zeros of a quadratic function by completing the square, you start by writing the quadratic function in the form $f(x) = ax^2 + bx + c$ . You then group the terms with the variable $x$ together, move the constant term to the right-hand side of the equation, add and subtract a constant term to the left-hand side of the equation, factor the perfect square trinomial, and write the function in the vertex form. You then set the function equal to zero and solve for $x$ .

Q: What are the steps to complete the square?

A: The steps to complete the square are:

Write the quadratic function in the form $f(x) = ax^2 + bx + c$ .
Group the terms with the variable $x$ together.
Move the constant term to the right-hand side of the equation.
Add and subtract a constant term to the left-hand side of the equation.
Factor the perfect square trinomial.
Write the function in the vertex form.

Q: How do I factor the perfect square trinomial?

A: To factor the perfect square trinomial, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the $x$ term. You then write the perfect square trinomial as the square of the binomial.

Q: What is the difference between completing the square and factoring?

A: Completing the square and factoring are two different techniques used to rewrite a quadratic function in a form that makes it easier to find the zeros. Factoring involves finding two binomials whose product is the quadratic function, while completing the square involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$ .

Q: When should I use completing the square?

A: You should use completing the square when the quadratic function cannot be factored easily, or when you need to find the zeros of the function.

Q: What are the advantages of completing the square?

A: The advantages of completing the square are:

It makes it easier to find the zeros of the function.
It makes it easier to graph the function.
It makes it easier to find the maximum or minimum value of the function.

Q: What are the disadvantages of completing the square?

A: The disadvantages of completing the square are:

It can be more difficult to complete the square than to factor the quadratic function.
It can be more difficult to find the zeros of the function using completing the square than using factoring.

Conclusion

In this article, we have discussed the technique of completing the square and how to use it to find the zeros of a quadratic function. We have also answered some common questions about completing the square and provided some tips and tricks for using this technique.