Problem 230 - Elementary Algebra

Question

Factor. \(64 x^{2}-96 x+36\)

Step-by-Step Solution

Answer

(8x - 6)^2

Step 1: Identify a common factor

Check if there is a common factor in all terms of the quadratic expression. In this case, there is no common factor other than 1.

Step 2: Recognize the square terms

Notice that the expression is a quadratic trinomial of the form \(ax^2 + bx + c\). Here, we have \(a = 64\), \(b = -96\), and \(c = 36\). Since the terms are perfect squares, the trinomial is a perfect square trinomial.

Step 3: Rewrite in the form of square of a binomial

A perfect square trinomial fits the form \((mx + n)^2\). First, identify \(m\) and \(n\):\[\sqrt{64x^2} = 8x\]\[\sqrt{36} = 6\].

Step 4: Verify by middle term adjustment

Adjust and verify the middle term. The middle term \(-96x\) should be equal to \(2 \times 8x \times 6\):\[-96x = 2 \times 8x \times -6\]. This checks out.

Step 5: Write the factored form

Thus, the original quadratic can be written in the form: \((8x - 6)^2\).

Key Concepts

quadratic expressions

Quadratic expressions are algebraic phrases of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. This means they do not change and \(x\) is the variable.
These expressions are called 'quadratic' because they involve the square of the variable () 'quad' means 'square or four-sided' in Latin).
For example, in the expression \(64x^2 - 96x + 36\), the term \(64x^2\) is the quadratic term, \(-96x\) is the linear term, and \(36\) is the constant term.
To factor quadratic expressions, different techniques can be used:

• Factoring out the greatest common factor (GCF)
• Using the quadratic formula
• Completing the square
• Identifying perfect square trinomials

. The task often depends on recognizing these forms and patterns in the expressions.

perfect square trinomials

Perfect square trinomials are a special case of quadratic expressions.
They take the form \((mx + n)^2 = m^2x^2 + 2mnx + n^2\).
Recognizing that an expression is a perfect square trinomial helps directly in factoring the expression.
This type of trinomial will have the first and last terms as perfect squares, and the middle term will be twice the product of the square roots of the first and third terms.
In the example \(64x^2 - 96x + 36\), \(64x^2\) is the square of \(8x\), and \(36\) is the square of \(6\).
The term \(-96x\) then matches \(2 \times 8x \times -6\).
Therefore, this trinomial can be factored into \((8x - 6)^2\).

factoring techniques

Factoring techniques are methods used to rewrite complex expressions as products of simpler factors.
One important technique is recognizing perfect square trinomials.
Here, the process involves:

• Identifying a common factor.
• Recognizing the structure of the trinomial.
• Expressing the trinomial as the square of a binomial.

Effective factoring can make solving quadratic equations faster and easier.
For example, knowing that \(64x^2 - 96x + 36\) is a perfect square trinomial allows for quickly writing \((8x - 6)^2\) without further steps.
Practicing these techniques sharpens algebra skills and reduces the risk of mistakes.