Question

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$64x^2 - 48x + 9 = 0$
$x = $
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Explanation:

Step1: Recognize the quadratic form

The equation \(64x^{2}-48x + 9 = 0\) is a quadratic equation. Notice that \(64x^{2}=(8x)^{2}\), \(9 = 3^{2}\), and \(- 48x=-2\times8x\times3\). So it fits the perfect - square trinomial formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a = 8x\) and \(b = 3\).

Step2: Factor the quadratic

Using the perfect - square trinomial formula, we can factor \(64x^{2}-48x + 9\) as \((8x - 3)^{2}=0\).

Step3: Solve for x

Set \(8x-3 = 0\). Add 3 to both sides of the equation: \(8x=3\). Then divide both sides by 8: \(x=\frac{3}{8}\). Since the square of a number is zero only when the number itself is zero, this is a repeated root.

Answer:

\(\frac{3}{8}\) (with multiplicity 2)