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Magazine Chapter 6: Problem 13
Factor. $$ 49 x 2+14 x+1 $$
Short Answer
Expert verified
The expression \(49x^2 + 14x + 1\) factors to \((7x + 1)^2\).
Step by step solution
01
Identify the Quadratic Expression
The problem is to factor the quadratic expression \(49x^2 + 14x + 1\). This is a trinomial of the form \(ax^2 + bx + c\) where \(a = 49\), \(b = 14\), and \(c = 1\).
02
Check for Perfect Square Trinomials
Notice that \(49 = 7^2\) and \(1 = 1^2\). There鈥檚 a possibility that this trinomial could be a perfect square, expressed as \((ux + v)^2\).
03
Square Root and Coefficient Analysis
Calculate the square root of \(49x^2\), which is \(7x\), and the square root of \(1\), which is \(1\). Check if twice the product of these square roots equals \(14x\), i.e., \(2 \cdot 7x \cdot 1 = 14x\), which matches the middle term.
04
Write as a Perfect Square
Since the conditions for a perfect square are satisfied, we can write the expression as \((7x + 1)^2\).
05
Expand to Verify
To ensure correctness, expand \((7x + 1)^2\) to verify it results back in \(49x^2 + 14x + 1\). Expanding gives \(49x^2 + 14x + 1\), confirming our factorization.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Perfect Square Trinomials
A perfect square trinomial is a special type of quadratic expression that can be expressed as the square of a binomial. In simple terms, it's a trinomial that comes from squaring a two-term binomial expression. This special form makes it easy to factor if you recognize the pattern.
- It follows the structure \((ax)^2 + 2abx + b^2\), which is the expansion of a binomial squared: \((ax + b)^2\).
- To identify a perfect square trinomial, check whether both the first and last terms are perfect squares.
- Then, make sure the middle term is twice the product of the square roots of the first and last terms.
Quadratic Trinomials
A quadratic trinomial is a polynomial with three terms, generally written in the form \(ax^2 + bx + c\). Each component of these expressions serves its own distinct purpose.
- The term \(ax^2\) is the quadratic term, indicating the degree of the expression, which influences the curve's shape when graphed.
- \(bx\) is the linear term, which affects the slope or tilt of the parabola.
- Finally, \(c\), the constant term, grounds the expression by dictating where the parabola will intersect the y-axis.
Expanding Binomials
Expanding a binomial refers to the multiplication of a two-term expression by itself or another binomial. This process can help in both verifying factorization and understanding the structure of polynomial expressions.
- The most common method of expanding a binomial is by using the distributive property, applying the formula \((a + b)^2 = a^2 + 2ab + b^2\).
- When expanding binomials, every term in the first binomial is multiplied by each term in the second one.
- This method is particularly useful when checking if a given factorization is correct by expanding to see if it matches the original expression.
