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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}
• 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