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Quadratic Equations Practice - 1

Quadratic Equation: 8x^2 + 15 = 26x

Solving by Factoring: Splitting the middle term
  • Multiply the number in first term with the number in third term: 8 * 15 = 120
  • Split the number in middle term, -26, into two parts whose product is 120: 
    • -26 = - 20 - 6, 
    • -20 * -6 = 120
  • Write -20x - 6x in place of -26x in the quadratic equation: 8x^2 - 20x - 6x + 15 = 0
  • Group the first two terms and next two terms by parenthesis: (8x^2 - 20x) - (6x + 15) = 0
  • Factor out common factors from each group: 4x(2x - 5) - 3(2x - 5) = 0
  • Factor out 2x - 5: (2x - 5)(4x - 3) = 0
  • By applying zero product rule, either (2x - 5) = 0 or (4x - 3) = 0. Solving both separately for x:
    • 2x - 5 = 0, x = 5/2
    • 4x - 3 = 0, x = 3/4
  • Answer is x = {5/2, 3/4}
Solving by Factoring: Completing the square
  • Write in standard form: 8x^2 - 26x + 15 = 0
  • Move the constant term (15) to right hand side:  8x^2 - 26x = -15
  • Divide throughout by coefficient of x^2: 
    • Dividing by 8: 8x^2/8 - 26x/8 = -15/8
    • Simplifying: x^2 - 13x/4 = -15/8
  • Take the coefficient of 'x', divide it by 2, then square it and add the resultant number on both sides of the equation:
    • Coefficient of x is -13/4
    • Dividing it by 2: -13/8
    • Squaring it:  (-13/8)^2
    • Add on both sides of the equation:  x^2 - 13x/4 +  (-13/8)^2  = -15/8 +  (-13/8)^2
    • Simplify right hand side: x^2 - 13x/4 +  (-13/8)^2  = 49/64
  • Compare the left hand side with a^2 - 2ab + b^2, so you get a = x and b = -13/8. Notice that -13x/4 = 2 * x * -13/8. Now applying the expansion formula a^2 - 2ab + b^2 = (a - b)^2, you can simplify the left hand side:  (x -13/8)^2  = 49/64
  • Solve for x:
    • Take square root on both sides: x - 13/8 = +/- sqrt(49/64) = +/- 7/8 (remember that the square root of a number can be either positive or negative)
    • Add 13/8 on both sides: x =  +/- 7/8 + 13/8
      • For positive (+) 7/8 , x = 7/8 + 13/8 = 20/8 = 5/2
      • For negative (-) 7/8 , x = -7/8 + 13/8 = -6/8 = 3/4
  • Answer is x = {5/2, 3/4}
Solving by the Quadratic Formula 
  • Write in standard form: 8x^2 - 26x + 15 = 0
  • Compare with general standard form of a quadratic equation ax^2 + bx + c = 0, 
    • a = 8
    • b = -26
    • c = 15
  • Write the quadratic formula: x = (-b +/- sqrt(b^2 - 4ac))/(2a)
  • Put values of a, b and c in the formula: x = (--26 +/- sqrt((-26)^2 - 4*8*15))/(2*8)
  • Simplify: x = (26 +/- 14)/16
    • For +14, x = (26 + 14)/16 = 5/2
    • For -14, x = (26 - 14)/16 = 3/4
Graphing the quadratic equation
  • Write in standard form:  y = 8x^2 - 26x + 15
  • Get the vertex coordinates:
    • Let the vertex be (h, k), then h = b/(2a), so h = -26/(2*8) =- 13/8
    • k is the value of the quadratic expression at x = h, so k = 8(13/8)^2 - 26*(13/8) + 15 = -49/8
    • The vertex coordinates are (13/8,  -49/8 ) or (1.625, -6.125)
  • Since this function has x-intercepts (not all quadratic functions have x-intercepts), no other points need to be calculated to graph it (of course, you can get a few other points to extend the graph). The x-intercepts are 5/2 and 3/4, whose coordinates are (5/2, 0) and (3/4, 0) or, in decimals, (2.5, 0) and (0.75, 0).
  • Plot the vertex and two x intercepts on a coordinate plane and join them with a free hand curve:
Graph of y = 8x^2 - 26x + 15