Verified Answer :

  • To find the equation with roots \(6\) and \(-6\), we use the factored form \((x - 6)(x + 6) = 0\).
  • Expanding this using the difference of squares identity, we get \(x^2 - 36 = 0\).
  • Comparing this to the given options, we identify that the correct equation is \(x^2 - 36 = 0\).
  • The correct option is \(\boxed{d}\).

Explanation

  1. Analysis of the Problem

We are asked to identify which of the four given quadratic equations has the solutions \(x_1 = 6\) and \(x_2 = -6\). A quadratic equation with roots \(r_1\) and \(r_2\) can be expressed in its factored form as \(a(x - r_1)(x - r_2) = 0\), where \(a\) is a non-zero constant. If we assume the simplest case where \(a = 1\), the equation becomes \((x - r_1)(x - r_2) = 0\).

  1. Determining the Correct Equation

Given the roots \(x_1 = 6\) and \(x_2 = -6\), we can substitute these values into the factored form:
\[(x - 6)(x - (-6)) = 0\]
Simplifying the expression within the second set of parentheses:
\[(x - 6)(x + 6) = 0\]
Applying the difference of squares identity, \((a - b)(a + b) = a^2 - b^2\), we expand the product:
\[x^2 - 6^2 = 0\]
\[x^2 - 36 = 0\]
Thus, the equation with roots \(6\) and \(-6\) is \(x^2 - 36 = 0\).

  1. Comparing with the Given Options

Now we compare our derived equation \(x^2 - 36 = 0\) with the given options:

A\(x^2 + 36 = 0\)
B\(x^2 - 12x + 36 = 0\)
C\(x^2 + 12x - 36 = 0\)
D\(x^2 - 36 = 0\)

The equation \(x^2 - 36 = 0\) corresponds exactly to option d. Therefore, this is the correct equation.

Examples

In physics, when modeling the motion of an object bouncing between two walls or reflecting off surfaces at equal distances from an origin, equations like \(x^2 - a^2 = 0\) frequently arise. For instance, if an object oscillates between position \(x = 6\) and \(x = -6\) on a coordinate axis, the points of reversal where the velocity is zero satisfy the equation \(x^2 - 36 = 0\). Solving such equations helps determine the boundaries or turning points of physical systems.

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