91Ó°ÊÓ

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). In this problem, we derived such an equation from the distance formula.
After squaring both sides to eliminate the square root, we got a quadratic in disguise: \[(9 - y)^2 = 119.\] We solved it by isolating the quadratic term, simplifying it, and then taking square roots. Remember, taking the square root of both sides of an equation typically yields two possible solutions, one positive and one negative. This is because \(\text{both} \ \text{positive and negative square roots} \ \text{of a number, squared, yield the same result}. \)

• Simplify and isolate the quadratic terms.
• Take the square root of both sides.
• Solve for the variable after taking both positive and negative roots.

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane can be computed using the distance formula. This formula derives from the Pythagorean theorem. The distance formula is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \] In this problem, we used:

• \(d = 12\),
  
  \(x_1 = 3\),

  \(y_1 = y\),

  \(x_2 = -2\),

  \(y_2 = 9\)

We substituted these values into the formula and solved for \(y\). This required algebraic manipulation, simplification, and solving a resulting quadratic equation. To solve such problems:

• Identify coordinates and distance.
• Substitute them into the distance formula.
• Solve the resulting equation step by step.

The Pythagorean theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the lengths of the other two sides \(a\) and \(b\): The distance formula \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] is derived from this theorem. Here's how:
Consider two points \((x_1, y_1)\) and \((x_2, y_2)\). Think of a line connecting these points as the hypotenuse of a right triangle, with the horizontal and vertical distances forming the legs. The lengths of these legs are \(x_2 - x_1\) and \(y_2 - y_1\). Applying the Pythagorean theorem gives us the distance formula. Using it, we found the distance between two points by:

• Computing horizontal and vertical differences.
• Applying the formula by summing their squares and taking the square root.