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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

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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

- The Quadratic Formula
- Problem Solving with the Quadratic Formula

Consider the General Quadratic EquationDerive 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

Now the Left-Hand-Side (LHS) is a PERFECT SquareDerive Quadratic Formula - 2

- Combine Terms inside the Radical over a Common Denom

- Take the Square Root of Both Sides

Note that Denom is, itself, a PERFECT SQDerive 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) 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) 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.

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) 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

- 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

- 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

- 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 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 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 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 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 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

- 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

- 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 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

- 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

TheQuadraticFormula

Graph y = |x|

- Make T-table

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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