Questions: Solve: 16 x^2 - 81 = 0 (A) -4/9, 4/9 (B) -81/16, 81/16 (C) -16/81, 16/81 (D) -9/4, 9/4

Solve: 16 x^2 - 81 = 0 (A) -4/9, 4/9 (B) -81/16, 81/16 (C) -16/81, 16/81 (D) -9/4, 9/4
Transcript text: Solve: $16 x^{2}-81=0$ (A) $-\frac{4}{9}, \frac{4}{9}$ (B) $-\frac{81}{16}, \frac{81}{16}$ (C) $-\frac{16}{81}, \frac{16}{81}$ D $-\frac{9}{4}, \frac{9}{4}$
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Solution

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Solution Steps

To solve the quadratic equation \(16x^2 - 81 = 0\), we can use the following steps:

  1. Recognize that the equation is in the form of \(a^2 - b^2 = 0\), which is a difference of squares.
  2. Factor the equation as \((4x - 9)(4x + 9) = 0\).
  3. Set each factor equal to zero and solve for \(x\).
Step 1: Recognize the Form of the Equation

The given equation is \(16x^2 - 81 = 0\). This is a difference of squares, which can be factored as \(a^2 - b^2 = (a - b)(a + b)\).

Step 2: Factor the Equation

We can rewrite \(16x^2 - 81\) as \((4x)^2 - 9^2\). Therefore, the equation can be factored as: \[ (4x - 9)(4x + 9) = 0 \]

Step 3: Solve for \(x\)

Set each factor equal to zero and solve for \(x\): \[ 4x - 9 = 0 \quad \text{or} \quad 4x + 9 = 0 \] Solving these equations, we get: \[ 4x - 9 = 0 \implies x = \frac{9}{4} \] \[ 4x + 9 = 0 \implies x = -\frac{9}{4} \]

Final Answer

The solutions to the equation \(16x^2 - 81 = 0\) are: \[ \boxed{x = \pm \frac{9}{4}} \] Thus, the correct answer is \(D\).

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