Problem 13 - Elementary Algebra

Question

Factor. $$ 49 x 2+14 x+1 $$

Step-by-Step Solution

Answer

The expression \(49x^2 + 14x + 1\) factors to \((7x + 1)^2\).

Step 1: 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 \(c = 1\).

Step 2: Check for Perfect Square Trinomials

Notice that \(49 = 7^2\) and \(1 = 1^2\). There’s a possibility that this trinomial could be a perfect square, expressed as \((ux + v)^2\).

Step 3: 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.

Step 4: Write as a Perfect Square

Since the conditions for a perfect square are satisfied, we can write the expression as \((7x + 1)^2\).

Step 5: 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.

Key Concepts

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.

For example, in the quadratic expression \(49x^2 + 14x + 1\), both \(49x^2\) and \(1\) are perfect squares, as we can write them as \((7x)^2\) and \((1)^2\) respectively. Since the middle term \(14x\) fits the pattern \(2 \times 7x \times 1\), the expression can be factored as \((7x + 1)^2\).

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.

Quadratic trinomials can often be factored into binomial expressions. By understanding the relationship between the terms, we can rearrange them and factor them more easily, identifying special forms like perfect square trinomials when applicable.

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.

For instance, expanding \((7x + 1)^2\) leads to \(49x^2 + 14x + 1\), affirming that \((7x + 1)^2\) is indeed the correct factorization of the original quadratic expression. This expansion process confirms the accuracy of the factorization, helping solidify understanding.