Verified Answer :

  • Identify the expression \(64x^2 - 48x + 9\) as a quadratic trinomial.
  • Recognize it as a perfect square trinomial of the form \((ax - b)^2 = a^2x^2 - 2abx + b^2\).
  • Match the terms \(a^2 = 64\), \(b^2 = 9\), and \(2ab = 48\) to confirm \(a=8\) and \(b=3\).
  • Write the final factored form: \(\boxed{(8x - 3)^2}\).

Explanation

  1. Analyzing the Quadratic Expression

The objective is to factor the quadratic expression \(64x^2 - 48x + 9\). A quadratic expression is in the form \(Ax^2 + Bx + C\), where in this case \(A = 64\), \(B = -48\), and \(C = 9\). A useful approach for such expressions is to check if they represent a perfect square trinomial, which follows the pattern \((px - q)^2 = p^2x^2 - 2pqx + q^2\).

  1. Testing for a Perfect Square Trinomial

To determine if the expression is a perfect square, we examine the first and last terms:

  • The first term is \(64x^2\), which is the square of \(8x\) since \((8x)^2 = 64x^2\).
  • The third term is \(9\), which is the square of \(3\) since \(3^2 = 9\).

Now, we compare the middle term of our expression (\(-48x\)) with the middle term of the expanded binomial \((8x - 3)^2\). The expansion of \((8x - 3)^2\) is \((8x)^2 - 2(8x)(3) + 3^2 = 64x^2 - 48x + 9\). Since our middle term \(-2(8x)(3) = -48x\) matches perfectly with the expression, we have confirmed that the expression is a perfect square trinomial.

  1. Factoring the Expression

Having confirmed the pattern, we can conclude that the expression \(64x^2 - 48x + 9\) is equal to the square of the binomial \((8x - 3)\). Thus, the fully factored form is \((8x - 3)^2\).

Examples

Factoring quadratic expressions is essential in fields like physics and engineering when determining the dimensions of physical structures or solving motion equations. For example, if the area of a square-shaped garden is given by the function \(A(x) = 64x^2 - 48x + 9\) square meters, factoring this into \((8x-3)^2\) allows you to quickly find that the side length of the garden is simply \(8x - 3\) meters. This simplifies calculations when trying to determine the perimeter or total fencing required for such a layout.

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