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Lesson 7 – Algebra of Quadratics – The Quadratic Formula
IB Math SL1 - Santowski
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Fast Five
Determine HOW many roots the following quadratic functions have:
(a) f(x) = x2 – 6x + 9 (b) f(x) = x2 – 6x + 5 (c) f(x) = x2 – 2x + 10 (d) f(x) = -½x2 + 2x + 4
(e) f(x) = x2 + 2x + c
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Lesson Objectives
Express a quadratic function in standard form and use the quadratic formula to find its zeros
Determine the number of real solutions for a quadratic equation by using the discriminant
Find and classify all roots of a quadratic equation
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BIG PICTURE
Sometimes the same function type can be written in a variety of different forms Sometimes the form of the equation can give us other important information
WHY?
Is there a connection between the form that the equation is written in and some of the key features of the graphs????
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(A) Skills Review
Given the following quadratic functions, determine the zeroes of the function using the QF (exact and rounded to 3 sf):
(a) f(x) = x2 – 6x + 6 (b) f(x) = 3x2 + 12x + 9 (c) g(x) = -¼x2 + 2x + 3
(d) EXPLAIN what you would predict the graphs of these quadratic functions to look like
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(A) Skills Review
The quadratic formula looks like:
a
acbbx
2
42
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(B) HOW Does the QF Work Examples
Solve 12x2 + 5x – 2 = 0 using the Q/F. Then rewrite the equation in factored form and in vertex form. PREDICT the appearance of the graph and its key features
Determine the roots of f(x) = 2x2 + x – 7 using the Q/F. Then rewrite the equation in factored form and in vertex form. PREDICT the appearance of the graph and its key features
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(B) HOW Does the QF Work Examples
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(B) HOW Does the QF Work Examples
Solve the system
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42
352
xy
xxy
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(B) HOW Does the QF Work Examples
Solve the system
42
352
xy
xxy
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(B) HOW Does the QF Work Examples Solve the equation and graphically verify the 2
solutions
Find the roots of 9(x – 3)2 – 16(x + 1)2 = 0
Solve 6(x – 1)2 – 5(x – 1)(x + 2) – 6(x + 2)2 = 0
1 11
3 1x x
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(B) HOW Does the QF Work Examples
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(C) WHY Does QF Work Solving Equations using C/S
Solve 0 = ax2 + bx + c by completing the square
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(C) WHY Does QF Work Solving Equations using C/S
If you solve 0 = ax2 + bx + c by completing the square, your solution should look familiar:
Which we know as the quadratic formula
a
acbbx
2
42
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(C) WHY Does QF Work Solving Equations using C/S
Here are some other rearrangements of the QF:
a
c
a
b
a
bx
a
acb
a
bx
a
acb
a
bx
2
2
2
2
22
4
4
2
2
4
2
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(C) WHY Does QF Work Examples
Given the quadratic function f(x) = x2 – 10x – 3, determine the distance between the roots and the axis of symmetry.
Determine the distance between the roots and the axis of symmetry of f(x) = 2x2 – 5x +1
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(D)WHY Do We Use Q/F The Discriminant Within the Q/F, the expression b2 – 4ac is referred to as
the discriminant
We can use the discriminant to classify the “nature of the roots” a quadratic function will have either 2 distinct, real roots, one real root, or no real roots this can be determined by finding the value of the discriminant
The discriminant will have one of 3 values: b2 – 4ac > 0 which means b2 – 4ac = 0 which means b2 – 4ac < 0 which means
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(D)WHY Do We Use Q/F The Discriminant Determine the value of the
discriminants in: (a) f(x) = x2 + 3x - 4 (b) f(x) = x2 + 3x + 2.25 (c) f(x) = x2 + 3x + 5
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(E)WHY Do We Use Q/F The Discriminant Based on the discriminant, indicate how many and
what type of solutions there would be given the following equations:
(a) 3x2 + x + 10 = 0 (b) x2 – 8x = -16 (c) 3x2 = -7x - 2
Verify your results using (i) an alternate algebraic method and (ii) graphically
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(D)WHY Do We Use Q/F The Discriminant Determine the value of W such that f(x) = Wx2 + 2x – 5 has one
real root. Verify your solution (i) graphically and (ii) using an alternative algebraic method.
Determine the value of b such that f(x) = 2x2 + bx – 8 has no solutions. Explain the significance of your results.
Determine the value of b such that f(x) = 2x2 + bx + 8 has no solutions.
Determine the value of c such that f(x) = f(x) = x2 + 4x + c has 2 distinct real roots.
Determine the value of c such that f(x) = f(x) = x2 + 4x + c has 2 distinct real rational roots.
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(E) Examples – Equation Writing and Forms of Quadratic Equations (1) Write the equation of the parabola that has
zeroes of –3 and 2 and passes through the point (4,5).
(2) Write the equation of the parabola that has a vertex at (4, −3) and passes through (2, −15).
(3) Write the equation of the parabola that has a y – intercept of –2 and passes through the points (1, 0) and (−2,12).
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(F) Homework
HW
Ex 8E, Q1acfghi; Q2abdef Ex 8H, Q5ghijkl Ex 8I.1, Q1bcd, Q2abc, Q3bcf Ex 8I.2, Q1cef, Q2ac, Q3