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$$$ax^2 = c$$, then $x = ext{±} rac{ ext{√}c}{ ext{√}a}$$$x = ext{±} rac{ ext{√}c}{ ext{√}a}$$.

Example Problems:

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

   • Step 1: Divide both sides by 2: $x^2 = 49$$$x^2 = 49$$.

   • Step 2: Taking the square root of both sides: $x = ext{±} ext{√}49$$$x = ext{±} ext{√}49$$.

   • Final Solutions: $x = 7$$$x = 7$$ or $x = -7$$$x = -7$$.

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

   • Step 1: Taking the square root of both sides: $y = ext{±} ext{√}81$$$y = ext{±} ext{√}81$$.

   • Final Solutions: $y = 9$$$y = 9$$ or $y = -9$$$y = -9$$.

Solving Quadratic Equations by Completing the Square

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

Example Problems:

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

   • Step 1: Move the constant to the other side: $x^2 + 6x = -4$$$x^2 + 6x = -4$$.

   • Step 2: Take half of the coefficient of x (which is 6), square it: $( rac{6}{2})^2 = 9$$$( rac{6}{2})^2 = 9$$.

   • Step 3: Add 9 to both sides: $x^2 + 6x + 9 = 5$$$x^2 + 6x + 9 = 5$$.

   • Step 4: Factor the left side: $(x + 3)^2 = 5$$$(x + 3)^2 = 5$$.

   • Step 5: Take the square root of both sides: $x + 3 = ext{±} ext{√}5$$$x + 3 = ext{±} ext{√}5$$.

   • Final Solutions: $x = -3 + ext{√}5$$$x = -3 + ext{√}5$$ or $x = -3 - ext{√}5$$$x = -3 - ext{√}5$$.

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

   • Step 1: Move the constant to the other side: $x^2 + 10x = -8$$$x^2 + 10x = -8$$.

   • Step 2: Take half of the coefficient of x (which is 10), square it: $( rac{10}{2})^2 = 25$$$( rac{10}{2})^2 = 25$$.

   • Step 3: Add 25 to both sides: $x^2 + 10x + 25 = 17$$$x^2 + 10x + 25 = 17$$.

   • Step 4: Factor the left side: $(x + 5)^2 = 17$$$(x + 5)^2 = 17$$.

   • Step 5: Take the square root of both sides: $x + 5 = ext{±} ext{√}17$$$x + 5 = ext{±} ext{√}17$$.

   • Final Solutions: $x = -5 + ext{√}17$$$x = -5 + ext{√}17$$ or $x = -5 - ext{√}17$$$x = -5 - ext{√}17$$.