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 $x = ext{±} \frac{ ext{√}c}{ ext{√}a}$.

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: $(\frac{6}{2})^2 = 9$.
   • 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: $(\frac{10}{2})^2 = 25$.
   • 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 $x = -5 - ext{√}17$.