Overview of Quadratic Polynomials and Their Zeroes

A quadratic polynomial is a polynomial of the second degree, which can be expressed in the form (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). The zeroes of a quadratic polynomial are the values of (x) that satisfy the equation (ax^2 bx c 0). In this article, we will discuss how to find the zeroes of a quadratic polynomial step-by-step using the example of the polynomial (4x^2 - 4x 1).

The quadratic polynomial in question is (4x^2 - 4x 1), and we will demonstrate how to find its zeroes using various methods.

Step-by-Step Solution

To solve the quadratic equation (4x^2 - 4x 1 0) and find its zeroes, we can follow these steps:

Step 1: Factorization

One of the most straightforward methods to find the zeroes of a quadratic polynomial is to factorize it. Let's start by recognizing that the polynomial can be written as a perfect square trinomial.

We can rewrite the polynomial as:

[ 4x^2 - 4x 1 (2x - 1)^2 ]

This means that the equation (4x^2 - 4x 1 0) is equivalent to:

[ (2x - 1)^2 0 ]

Setting the factor inside the parentheses equal to zero gives us:

[ 2x - 1 0 ]

Solving for (x), we get:

[ 2x 1 ]

[ x frac{1}{2} ]

Therefore, (x frac{1}{2}) is the zero of the quadratic polynomial, and it is a repeated zero because it satisfies the equation twice.

Step 2: Using the Quadratic Formula

Another method to find the zeroes of a quadratic polynomial is to use the quadratic formula. The quadratic formula is given by:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

For the quadratic polynomial (4x^2 - 4x 1), we have:

[ a 4, b -4, c 1 ]

Plugging these values into the quadratic formula, we get:

[ x frac{-(-4) pm sqrt{(-4)^2 - 4(4)(1)}}{2(4)} ]

[ x frac{4 pm sqrt{16 - 16}}{8} ]

[ x frac{4 pm sqrt{0}}{8} ]

[ x frac{4}{8} ]

[ x frac{1}{2} ]

Again, we find that (x frac{1}{2}) is the zero of the quadratic polynomial, and it is a repeated zero.

Conclusion

Through both factorization and the quadratic formula, we have determined that the zero of the quadratic polynomial (4x^2 - 4x 1) is (x frac{1}{2}), which is a repeated zero. This example demonstrates the importance of understanding different methods to solve quadratic equations.

Additional Tips for Solving Quadratic Equations

Factorization: Always look for a way to factorize the polynomial first. This can often be the quickest and most straightforward method. Quadratic Formula: When factorization is not possible, the quadratic formula is a reliable method to find the zeroes. Practice: The more you practice solving quadratic equations, the more comfortable you will become with different methods and the easier it will be to choose the most appropriate one for each problem.

Now that you understand the concept of zeroes in quadratic polynomials and how to find them, you can apply these methods to other quadratic equations. Happy solving!