Understanding Quadratic Equations:

- A quadratic equation is a polynomial equation of degree 2. It can be expressed in the general form:
ax² + bx + c = 0
Methods for Solving Quadratic Equations:
-
Factoring:
- Factor the quadratic expression into two linear factors.
- Set each factor equal to zero and solve for x.
- Example:
x² - 5x + 6 = 0 (x - 2)(x - 3) = 0 x = 2 or x = 3
-
Completing the Square:
- Convert the quadratic equation into a perfect square trinomial.
- Take the square root of both sides and solve for x.
- Example:
x² - 6x + 2 = 0 x² - 6x + 9 = 7 (x - 3)² = 7 x - 3 = ±√7 x = 3 ± √7
-
Quadratic Formula:
- Apply the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
- Substitute the values of a, b, and c from the quadratic equation.
- Simplify and solve for x.
- Apply the quadratic formula:
Example:
Solve the equation: 2x² - 3x - 5 = 0
Method 1: Factoring
- We cannot factor this equation into two linear factors.
Method 2: Completing the Square
- Divide the equation by 2 to simplify:
x² - (3/2)x - (5/2) = 0
- Add (3/4)² to both sides to complete the square:
x² - (3/2)x + (9/16) = (9/16) + (5/2) (x - 3/4)² = 49/16 x - 3/4 = ±7/4 x = 3/4 ± 7/4 x = 5/2 or x = -1
Method 3: Quadratic Formula
- Substitute a = 2, b = -3, and c = -5 into the formula:
x = (-(-3) ± √((-3)² - 4(2)(-5))) / (2(2)) x = (3 ± √49) / 4 x = (3 ± 7) / 4 x = 5/2 or x = -1
Choosing the Best Method:
- Factoring is often the quickest method if it's possible.
- Completing the square is useful when the quadratic equation is not easily factored.
- The quadratic formula is always applicable, even if factoring or completing the square is difficult.
I hope this explanation helps! Feel free to ask if you have any more questions.


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