Verified Answer :

  • Use the factored form of a quadratic equation \((x - r_1)(x - r_2) = 0\) with given roots \(6\) and \(-6\).
  • Apply the difference of squares identity to expand \((x - 6)(x + 6) = x^2 - 36\).
  • Alternatively, verify using Vieta's formulas, where the sum of roots is \(0\) and the product is \(-36\), resulting in the equation \(x^2 - 36 = 0\).
  • Conclude that the correct choice is option d, \(\boxed{d}\).

Explanation

  1. Deriving the Equation from Roots

To identify the correct quadratic equation with roots \(x_1 = 6\) and \(x_2 = -6\), we can use the fundamental relationship between the roots of a quadratic equation and its coefficients. A quadratic equation with roots \(r_1\) and \(r_2\) can be expressed in its factored form as:
\[a(x - r_1)(x - r_2) = 0\]
For our specific roots \(r_1 = 6\) and \(r_2 = -6\), the equation becomes:
\[(x - 6)(x - (-6)) = (x - 6)(x + 6) = 0\]
Expanding this expression using the difference of squares identity, \((A - B)(A + B) = A^2 - B^2\), we get:
\[x^2 - 6^2 = x^2 - 36 = 0\]
Thus, the equation is \(x^2 - 36 = 0\).

  1. Verification using Vieta's Formulas

Alternatively, we can use Vieta's formulas. For a quadratic equation of the form \(x^2 + Bx + C = 0\), the sum of the roots is \(-B\) and the product of the roots is \(C\).
Given the roots \(x_1 = 6\) and \(x_2 = -6\):
Sum of the roots: \(r_1 + r_2 = 6 + (-6) = 0\). Therefore, \(-B = 0\), which means \(B = 0\).
Product of the roots: \(r_1 \times r_2 = 6 \times (-6) = -36\). Therefore, \(C = -36\).
Substituting these coefficients into the general form \(x^2 + Bx + C = 0\), we obtain:
\[x^2 + 0x - 36 = 0 \implies x^2 - 36 = 0\]
This confirms our previous result. Comparing this with the provided options:

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

The correct option is (d).

Examples

In engineering and physics, solving quadratic equations is essential for determining equilibrium points or boundaries. For instance, if you are designing a suspension bridge and need to find the points where a support cable crosses a specific horizontal axis, these points are often roots of a quadratic function describing the cable's arc. Knowing how to relate the roots to the equation helps in modeling and building stable structures.

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