Problem 6 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ 0=x^{2}-4 x+4 $$

Step-by-Step Solution

Answer

The solution to the equation \(x^{2} -4x + 4 = 0\) is \( x = 2 \).

Step 1: Identify a, b and c

From the equation \(0 = x^{2} -4x + 4\), it can be identified that \(a = 1\), \(b = -4\), and \(c = 4\).

Step 2: Factoring the equation

The equation needs to be factored in the form \(0 = (x - N1)(x - N2)\), where \(N1 \times N2 = a \times c\) and \(N1 + N2 = b\). After trying possible factors, we can deduce that \(N1 = N2 = 2\). Thus the factored form of the equation is \(0 = (x - 2)^{2}\).

Step 3: Solving for x

To solve for x, expand the factored equation to \(0 = x^{2} - 4x + 4\). Since this is a quadratic equation, it can be solved by setting the factors equal to zero. As \(x - 2 = 0\), x equals 2.

Key Concepts

Factoring

Factoring is one of the most appealing methods to solve quadratic equations when it is applicable. It involves rewriting the quadratic equation in a form where it is expressed as a product of binomials. This method is straightforward when the equation easily factors into integers.

To factor a quadratic equation such as \(0 = x^2 - 4x + 4\), you should begin by identifying the coefficients: \(a = 1\), \(b = -4\), and \(c = 4\). The goal is to express the quadratic in the form \((x - N1)(x - N2)\), where:

• \(N1 \times N2 = a \times c\)
• \(N1 + N2 = b\)

For this specific equation, since \(N1 = N2 = 2\), the factored form is \((x - 2)^2\). Now, this form is significant because it implies the equation has a perfect square, leading us to a solution where the roots are conveniently visible.

Quadratic Formula

When factoring is not directly possible, the quadratic formula is a reliable alternative for solving any quadratic equation. This formula is derived from the standard quadratic form \(ax^2 + bx + c = 0\), providing a universal solution.

The quadratic formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This method includes calculating the determinant (\(b^2 - 4ac\)), often called the "discriminant," which indicates the nature of the roots:

• If positive, the equation has two distinct real roots.
• If zero, there's exactly one real root (a repeated root).
• If negative, the roots are complex or non-real.

For the equation \(0 = x^2 - 4x + 4\), using the formula confirms the roots as \(x = 2\), demonstrating how it aligns perfectly with our previous factoring outcome, thereby ensuring both accuracy and reliability.

Roots of a Quadratic Equation

The roots or solutions to a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots are the x-intercepts of the quadratic's graph, and they can be found using various methods such as factoring, the quadratic formula, or completing the square.

In our given problem \(x^2 - 4x + 4 = 0\), both the factoring and quadratic formula methods show that the root is \(x = 2\). It's significant to note that this equation's root is repetitive, that is, \(x - 2\) appears twice in the factored form \((x - 2)^2 = 0\). Such a root is known as a "double root," indicating the parabola touches the x-axis at \(x = 2\).

• Understanding roots helps in graphing, determining the parabola's vertex, and further analyzing the function's behavior.
• Recognizing the multiplicity of roots is essential for interpreting and predicting the quadratic's graphical representation.

Ultimately, mastering the concept of finding roots in different contexts lays the groundwork for more advanced mathematical problem-solving.