Quadratic Equations and Methods

Quadratic Equations

Solving Quadratic Equations Using the Square Root Property

• Definition: The square root property states that if ax^2 = c, then x = ext{±} rac{ ext{√}c}{ ext{√}a}.

Example Problems:

1. Problem (a): Solve the equation 2x^2 = 98.

   • Step 1: Divide both sides by 2: x^2 = 49.
   • Step 2: Taking the square root of both sides: x = ext{±} ext{√}49.
   • Final Solutions: x = 7 or x = -7.

2. Problem (b): Solve the equation y^2 = 81.

   • Step 1: Taking the square root of both sides: y = ext{±} ext{√}81.
   • Final Solutions: y = 9 or y = -9.

Solving Quadratic Equations by Completing the Square

• Definition: Completing the square involves rewriting a quadratic equation in the form a(x-h)^2 = k.

Example Problems:

1. Problem (a): Solve the equation x^2 + 6x + 4 = 0.

   • Step 1: Move the constant to the other side: x^2 + 6x = -4.
   • Step 2: Take half of the coefficient of x (which is 6), square it: (rac{6}{2})^2 = 9.
   • Step 3: Add 9 to both sides: x^2 + 6x + 9 = 5.
   • Step 4: Factor the left side: (x + 3)^2 = 5.
   • Step 5: Take the square root of both sides: x + 3 = ext{±} ext{√}5.
   • Final Solutions: x = -3 + ext{√}5 or x = -3 - ext{√}5.

2. Problem (b): Solve the equation x^2 + 10x + 8 = 0.

   • Step 1: Move the constant to the other side: x^2 + 10x = -8.
   • Step 2: Take half of the coefficient of x (which is 10), square it: (rac{10}{2})^2 = 25.
   • Step 3: Add 25 to both sides: x^2 + 10x + 25 = 17.
   • Step 4: Factor the left side: (x + 5)^2 = 17.
   • Step 5: Take the square root of both sides: x + 5 = ext{±} ext{√}17.
   • Final Solutions: x = -5 + ext{√}17 or x = -5 - ext{√}17.