Quadratic Equations and Methods

Quadratic Equations

Solving Quadratic Equations Using the Square Root Property
  • Definition: The square root property states that if ax2=cax^2 = c, then x = ext{±} rac{ ext{√}c}{ ext{√}a}.
Example Problems:

  1. Problem (a): Solve the equation 2x2=982x^2 = 98.

    • Step 1: Divide both sides by 2: x2=49x^2 = 49.
    • Step 2: Taking the square root of both sides: x=ext±ext√49x = ext{±} ext{√}49.
    • Final Solutions: x=7x = 7 or x=−7x = -7.

  2. Problem (b): Solve the equation y2=81y^2 = 81.

    • Step 1: Taking the square root of both sides: y=ext±ext√81y = ext{±} ext{√}81.
    • Final Solutions: y=9y = 9 or y=−9y = -9.
Solving Quadratic Equations by Completing the Square
  • Definition: Completing the square involves rewriting a quadratic equation in the form a(x−h)2=ka(x-h)^2 = k.
Example Problems:

  1. Problem (a): Solve the equation x2+6x+4=0x^2 + 6x + 4 = 0.

    • Step 1: Move the constant to the other side: x2+6x=−4x^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: x2+6x+9=5x^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√5x + 3 = ext{±} ext{√}5.
    • Final Solutions: x=−3+ext√5x = -3 + ext{√}5 or x=−3−ext√5x = -3 - ext{√}5.

  2. Problem (b): Solve the equation x2+10x+8=0x^2 + 10x + 8 = 0.

    • Step 1: Move the constant to the other side: x2+10x=−8x^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: x2+10x+25=17x^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√17x + 5 = ext{±} ext{√}17.
    • Final Solutions: x=−5+ext√17x = -5 + ext{√}17 or x=−5−ext√17x = -5 - ext{√}17.