Pre cal 1.5

Solving Quadratic Equations

Introduction to Quadratic Equations

  • Quadratic equations can often be solved using various methods, primarily including factoring, using the square root property, and completing the square.

Square Root Property

  • To apply the square root property, you rearrange the quadratic equation so that one side equals a perfect square.

  • For example, rearranging:

    • Start with: 9x² = -25

    • Move 25 to the other side: 9x² = -25

    • Divide both sides by 9: x² = -25/9

Square Roots of Negative Numbers

  • Taking the square root of a negative number involves imaginary numbers:

    • √-25 = √(25 × -1) = 5i

  • This leads to: 3x = ±5i.

  • To isolate x, divide by 3: x = ±(5i/3).

Combining and Splitting Expressions

  • When combining expressions, ensure radicals are like terms:

    • For instance, combining terms like -3 + 2√7 does not work because one term includes a square root while the other is a constant.

Completing the Square Method

  • Completing the square is used when solving quadratic equations, particularly those that cannot be easily factored.

  • Process overview:

    1. Move the constant term to the opposite side of the equation to isolate the x-terms.

    2. Calculate a specific quantity to add to both sides:

    • For the equation x² + 8x + 14 = 0, first isolate the constant:

    • x² + 8x = -14

    • Calculate: (8/2)² = 16, then add 16 to both sides.

Resulting Equation

  • The left side becomes a perfect square: (x + 4)² = 2 after combining terms on the right side: -14 + 16 = 2.

  • Taking the square root leads to applying the square root property:

    • x + 4 = ±√2, hence x = -4 ± √2.

Summary of Steps

  1. Rearrange to isolate the quadratic expression.

  2. Calculate (coefficient/2)² to find the value to add.

  3. Adjust both sides of the equation accordingly.

  4. Factor the left side and simplify the right side.

  5. Apply the square root property to find x values.

  6. Verify the directions on the problem to ensure appropriate solving method.

Practice Problems

  • Students are encouraged to practice problems 1 through 13 related to quadratic equations, utilizing completing the square method for understanding.

Additional Points

  • Note that different quadratic equations may require distinct approaches, so familiarize with each method thoroughly.

  • Understanding that completing the square method is foundational, as it relates to higher-level mathematics (e.g., graphing circles and parabolas) is crucial for long-term learning.