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Chabot Mathematics. §8.2 Quadratic Equation. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] MTH 55. 8.1. Review §. Any QUESTIONS About §8.1 → Complete the Square Any QUESTIONS About HomeWork §8.1 → HW-37. The Quadratic Formula.

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Chabot Mathematics

§8.2 QuadraticEquation

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


Review

MTH 55

8.1

Review §

  • Any QUESTIONS About

    • §8.1 → Complete the Square

  • Any QUESTIONS About HomeWork

    • §8.1 → HW-37


The quadratic formula
The Quadratic Formula

  • The solutions of ax2 + bx + c = 0 are given by

This is one of theMOST FAMOUSFormulas in allof Mathematics


8 2 quadratic formula
§8.2 Quadratic Formula

  • The Quadratic Formula

  • Problem Solving with the Quadratic Formula


Derive quadratic formula 1

Consider the General Quadratic Equation

Derive Quadratic Formula - 1

  • Next, Divide by “a” to give the second degree term the coefficient of 1

  • Where a, b, c are CONSTANTS

  • Solve This Eqn for x by Completing the Square

  • First; isolate the Terms involving x

    • Now add to both Sides of the eqn a “quadratic supplement” of (b/2a)2


    Derive quadratic formula 2

    Now the Left-Hand-Side (LHS) is a PERFECT Square

    Derive Quadratic Formula - 2

    • Combine Terms inside the Radical over a Common Denom

    • Take the Square Root of Both Sides


    Derive quadratic formula 4

    Note that Denom is, itself, a PERFECT SQ

    Derive Quadratic Formula - 4

    • Now Combine over Common Denom

    • But this the Renowned QUADRATIC FORMULA

    • Note That it was DERIVED by COMPLETING theSQUARE

    • Next, Isolate x


    Example a 2 x 2 9 x 5 0
    Example a) 2x2 + 9x− 5 = 0

    • Solve using the Quadratic Formula: 2x2 + 9x− 5 = 0

    • Soln a) Identify a, b, and c and substitute into the quadratic formula:

    2x2 + 9x−5 = 0

    a bc

    • Now Know a, b, and c


    Solution a 2 x 2 9 x 5 0
    Solution a) 2x2 + 9x− 5 = 0

    • Using a = 2, b = 9, c = −5

    Recall the Quadratic Formula→ Sub for a, b, and c

    Be sure to write the fraction bar ALL the way across.


    Solution a 2 x 2 9 x 5 01
    Solution a) 2x2 + 9x− 5 = 0

    • From Last Slide:

    • So:

    • The Solns:


    Example b x 2 12 x 4
    Example b) x2 = −12x + 4

    • Soln b) write x2 = −12x + 4 in standard form, identify a, b, & c, and solve using the quadratic formula:

      1x2 + 12x–4 = 0

    a bc


    Example c 5 x 2 x 3 0
    Example c) 5x2−x + 3 = 0

    • Soln c) Recognize a = 5, b = −1, c = 3 → Sub into Quadratic Formula

    • The COMPLEX No. Soln

    Since the radicand, –59, is negative, there are NO real-number solutions.


    Quadratic equation graph

    vertex

    Quadratic Equation Graph

    • The graph of a quadratic eqn describes a “parabola” which has one of a:

      • Bowl shape

      • Dome shape

    x intercepts

    • The graph, dependingon the “Vertex” Location,may have different numbers of of x-intercepts: 2 (shown), 1, or NONE


    The discriminant
    The Discriminant

    • It is sometimes enough to know what type of number (Real or Complex) a solution will be, without actually solving the equation.

    • From the quadratic formula, b2 – 4ac, is known as the discriminant.

    • The discriminant determines what type of number the solutions of a quadratic equation are.

      • The cases are summarized on the next sld



    Example discriminant
    Example  Discriminant

    • Determine the nature of the solutions of:

      5x2− 10x + 5 = 0

    • SOLUTION

    • Recognize a = 5, b = −10, c = 5

    • Calculate the Discriminant

      b2− 4ac = (−10)2− 4(5)(5) = 100 − 100 = 0

    • There is exactly one, real solution.

      • This indicates that 5x2− 10x + 5 = 0 can be solved by factoring  5(x− 1)2 = 0


    Example discriminant1
    Example  Discriminant

    • Determine the nature of the solutions of:

      5x2− 10x + 5 = 0

    • SOLUTION Examine Graph

    • Notice that the Graphcrosses the x-axis (where y = 0) atexactly ONE point aspredicted by the discriminant


    Example discriminant2
    Example  Discriminant

    • Determine the nature of the solutions of:

      4x2−x + 1 = 0

    • SOLUTION

    • Recognize a = 4, b = −1, c = 1

    • Calculate the Discriminant

      b2 – 4ac = (−1)2− 4(4)(1) =1 − 16 = −15

    • Since the discriminant is negative, there are two NONreal complex-number solutions


    Example discriminant3
    Example  Discriminant

    • Determine the nature of the solutions of:

      4x2− 1x + 1 = 0

    • SOLUTION Examine Graph

    • Notice that the Graphdoes NOT cross the x-axis (where y = 0) indicating that there are NO real values for x that satisfy this Quadratic Eqn


    Example discriminant4
    Example  Discriminant

    • Determine the nature of the solutions of:

      2x2 + 5x = −1

    • SOLUTION: First write the eqn in Std form of ax2 + bx + c = 0 →

      2x2 + 5x + 1 = 0

    • Recognize a = 2, b = 5, c = 1

    • Calculate the Discriminant

      b2 – 4ac = (5)2 – 4(2)(1) = 25 – 8 = 17

    • There are two, real solutions


    Example discriminant5
    Example  Discriminant

    • Determine the nature of the solutions of:

      0.3x2− 0.4x + 0.8 = 0

    • SOLUTION

    • Recognize a = 0.3, b = −0.4, c = 0.8

    • Calculate the Discriminant

      b2− 4ac = (−0.4)2− 4(0.3)(0.8) =0.16–0.96 = −0.8

    • Since the discriminant is negative, there are two NONreal complex-number solutions


    Writing equations from solns
    Writing Equations from Solns

    • The principle of zero products informs that this factored equation (x − 1)(x + 4) = 0 has solutions1 and −4.

    • If we know the solutions of an equation, we can write an equation, using the principle of Zero Products in REVERSE.


    Example write eqn from solns
    Example  Write Eqn from solns

    • Find an eqn for which 5 & −4/3 are solns

    • SOLUTION

    x = 5 orx = –4/3

    x – 5 = 0 orx + 4/3 = 0

    Get 0’s on one side

    Using the principle of zero products

    (x – 5)(x + 4/3) = 0

    x2 – 5x + 4/3x – 20/3 = 0

    Multiplying

    3x2 – 11x – 20 = 0

    Combining like terms and clearing fractions


    Example write eqn from solns1
    Example  Write Eqn from solns

    • Find an eqn for which 3i & −3i are solns

    • SOLUTION

    x = 3iorx = –3i

    x – 3i = 0 orx + 3i = 0

    Get 0’s on one side

    Using the principle of zero products

    (x – 3i)(x + 3i) = 0

    x2 – 3ix + 3ix – 9i2= 0

    Multiplying

    x2+ 9 = 0

    Combining like terms


    Whiteboard work
    WhiteBoard Work

    • Problems From §8.2 Exercise Set

      • 18, 30, 44, 58

    Solving Quadratic Equations

    1. Check to see if it is in the formax2 = p or (x + c)2 = d.

    • If it is, use the square root property

      2. If it is not in the form of (1), write it in standard form:

    • ax2 + bx + c = 0 with a and b nonzero.

      3. Then try factoring.

      4. If it is not possible to factor or if factoring seems difficult, use the quadratic formula.

    • The solns of a quadratic eqn cannot always be found by factoring. They can always be found using the quadratic formula.


    All done for today
    All Done for Today

    TheQuadraticFormula


    Chabot Mathematics

    Appendix

    Bruce Mayer, PE

    Licensed Electrical & Mechanical [email protected]


    Graph y x
    Graph y = |x|

    • Make T-table


    Quadratic equation graph1
    Quadratic Equation Graph

    • The graph of a quadratic eqn describes a “parabola” which has one of a:

      • Bowl shape

      • Dome shape

    • The graph, dependingon the “Vertex” Locationmay have different numbers of x-intercepts: 2 (shown), 1, or NONE


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