Problem 24 - Elementary Algebra

Question

Factor completely. $$ 36 x 2-12 x+1 $$

Step-by-Step Solution

Answer

The expression factors as \((6x - 1)^2\).

Step 1: Identify the Quadratic

The given expression is a quadratic trinomial, which follows the standard format \(ax^2 + bx + c\). In this case, \(a = 36\), \(b = -12\), and \(c = 1\).

Step 2: Check for a Perfect Square Trinomial

A perfect square trinomial is one that can be expressed as \((px + q)^2 = p^2x^2 + 2pqx + q^2\). Here, we check if \(36x^2-12x+1\) can be rewritten as \((px + q)^2\).

Step 3: Verify the Perfect Square Conditions

Calculate the square root of the first term: \(\sqrt{36} = 6\). Calculate the square root of the third term: \(\sqrt{1} = 1\). Check if twice the product of these square roots gives the middle term: \(2 \times 6 \times 1 = 12\), which is indeed the absolute value of the middle coefficient (-12).

Step 4: Solve for the Perfect Square Binomial

Since the conditions for a perfect square are satisfied, the expression can be written as a binomial square: \((6x - 1)^2\). This means \(36x^2 - 12x + 1 = (6x - 1)(6x - 1)\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression that can be rewritten as the square of a binomial. This occurs when the quadratic expression fits the form \((px + q)^2 = p^2x^2 + 2pqx + q^2\). To determine if a quadratic is a perfect square trinomial, you need to:

• Find the square roots of the first and last terms.
• Ensure that double the product of these square roots is equal to the middle term.

In the example \(36x^2 - 12x + 1\), the square roots of the first and third terms are \(6\) and \(1\) respectively. The middle term \(-12\) is indeed twice the product of \(6\) and \(1\) when considering the negative sign. When these conditions are met, you have a perfect square trinomial that can re-factor into a binomial square.

Quadratic Equation

A quadratic equation is one that can be expressed in the form \(ax^2 + bx + c = 0\). In our exercise, we start from a quadratic trinomial, \(36x^2 - 12x + 1\), which is not equal to zero, but can be worked with similarly because we are only factoring.Quadratic equations are often solved using different methods:

• Factoring, when the equation can be rewritten as a product of binomials.
• The quadratic formula, when factoring is not straightforward.
• Completing the square, another technique that can also highlight the concept of binomial squares.

Understanding how to manipulate the equation to get into one of these forms is critical for solving quadratics effectively.

Binomial Square

When a quadratic expression is rewritten as a binomial square, it takes the form \((px + q)(px + q)\) or simply \((px + q)^2\). This form makes it easy to see the structure of the perfect square trinomial it originated from. The expression \(36x^2 - 12x + 1\) can be refactored into \((6x - 1)^2\).Recognizing a binomial square involves:

• Identifying the "p" and "q" from the square roots of the first and last terms of the trinomial.
• Checking that the middle term is twice the product of these terms.

The advantage of writing quadratic expressions as binomial squares is that they can be easily expanded or solved, making them very useful in algebra.