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Magazine Chapter 7: Problem 43
As \((x+b)^{2}+C\) or \(-(x-b)^{2}+C\) by completing the square. $$ x^{2}-4 x+8 $$
Short Answer
Expert verified
The expression can be written as \((x-2)^2 + 4\).
Step by step solution
01
Identify the Quadratic Expression Format
The expression given is in the form of \(ax^2 + bx + c\), where \(a = 1\), \(b = -4\), and \(c = 8\). This indicates that we can apply the completing the square method.
02
Find the Term to Complete the Square
To complete the square, we need to add and subtract a term inside the expression. The term is found using \( \left(\frac{b}{2}\right)^2 \). For this expression, \( b = -4 \), so: \[ \left(\frac{-4}{2}\right)^2 = 4 \].
03
Complete the Square
Rewrite the original expression by adding and subtracting the term found in Step 2: \[ x^2 - 4x + 8 = (x^2 - 4x + 4) + 4 \].
04
Transform to Square Form
Rewrite \(x^2 - 4x + 4\) as \((x - 2)^2\) since \((x - 2)^2 = x^2 - 4x + 4\). Thus, the expression becomes: \[ (x - 2)^2 + 4 \].
05
Choose the Correct Form
The rewritten expression \((x - 2)^2 + 4\) is in the form \((x+b)^2 + C\), where \(b = -2\) and \(C = 4\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Expression
A quadratic expression is a type of polynomial expression with a degree of two. What this means is that the highest power of the variable, usually denoted as \(x\), is squared. A standard quadratic expression comes in the form of \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, and \(a\) cannot be zero.
Let's break it down:
Let's break it down:
- The term \(ax^2\) is known as the quadratic term, and it's the form that tells us the expression is a quadratic.
- The term \(bx\) is the linear term, as its highest power of \(x\) is one.
- The constant \(c\) is a standalone number without a variable part.
- \(a = 1\), meaning the squared term is just \(x^2\).
- \(b = -4\), involving the linear term \(-4x\).
- \(c = 8\), which is the constant part.
Square Form Transformation
Square form transformation is a technique used with quadratic expressions to rewrite them as a perfect square trinomial plus a constant. This is commonly used in solving quadratic equations and analyzing their graphs. The method of completing the square is pivotal to achieve this transformation.
To transform a quadratic expression, follow these steps:
To transform a quadratic expression, follow these steps:
- Identify the coefficient of the linear term \(b\) from the expression \(ax^2 + bx + c\).
- Calculate \( \left(\frac{b}{2}\right)^2 \). This value is called the "square completion term." For example, if \(b = -4\), then \( \left(\frac{-4}{2}\right)^2 = 4 \).
- Add and subtract this square completion term within the expression to form a perfect square trinomial, \((x + d)^2\).
- The quadratic expression is rewritten as \((x^2 - 4x + 4) + 4\).
- Recognize \((x - 2)^2\) as the squared term.
Mathematical Problem Solving
Mathematical problem solving involves applying concepts and techniques to simplify, rearrange, or solve mathematical expressions or equations. Here, completing the square provides a method to transform and solve quadratic expressions effectively.
Key steps in mathematical problem solving with quadratics include:
Key steps in mathematical problem solving with quadratics include:
- Understanding the nature of the quadratic expression: What makes it quadratic and what are its coefficients?
- Applying appropriate techniques: For example, using completing the square for transformation.
- Rechecking solutions: Does the transformed expression maintain its mathematical properties?
- Identifying the expression \(x^2 - 4x + 8\) as a quadratic.
- Rewriting it by completing the square to achieve \((x - 2)^2 + 4\).
- Recognizing that the form \((x + b)^2 + C\) is correctly achieved, ensuring any problem solver or student fully comprehends the transformation process.
