solve 4x squared minus 4x plus 1 equals 0 using the quadratic formula

To solve the quadratic equation \(4x^2 - 4x + 1 = 0\) using the quadratic formula, follow these steps:

### Step 1: Identify the coefficients
The quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 4\)
- \(b = -4\)
- \(c = 1\)

### Step 2: Write down the quadratic formula
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step 3: Substitute the values of \(a\), \(b\), and \(c\) into the formula
\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)}
\]

### Step 4: Simplify inside the square root
First, calculate \(b^2 - 4ac\):
\[
b^2 - 4ac = (-4)^2 - 4(4)(1) = 16 - 16 = 0
\]

So the expression becomes:
\[
x = \frac{4 \pm \sqrt{0}}{8}
\]

### Step 5: Simplify the square root
\(\sqrt{0} = 0\), so:
\[
x = \frac{4 \pm 0}{8} = \frac{4}{8} = \frac{1}{2}
\]

### Final Answer:
The solution to the equation \(4x^2 - 4x + 1 = 0\) is:
\[
x = \frac{1}{2}
\]
Since the discriminant is zero, there is only one real root, and it is a repeated root.

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Solve \(x^2 - 2x + 1 = 0\) using the quadratic formula.
2. What does it mean when the discriminant of a quadratic equation is zero?
3. Solve \(3x^2 + 6x + 3 = 0\) and identify the nature of the roots.
4. Find the roots of the equation \(2x^2 + 4x + 2 = 0\).
5. How can you factorize \(4x^2 - 4x + 1\) directly without using the quadratic formula?

**Tip:** When the discriminant (\(b^2 - 4ac\)) is zero, the quadratic equation has exactly one real root, meaning the parabola touches the x-axis at a single point.

Learn how to solve the quadratic equation 4x^2 - 4x + 1 = 0 using the quadratic formula with a step-by-step solution. This problem involves identifying coefficients, applying the quadratic formula, and understanding the nature of roots. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation