Question

Quadratic Equations question Will give fan and medal!

Answer 1

A quadratic equation is shown below:

9x^2 - 16x + 60 = 0

Part A: Describe the solution(s) to the equation by just determining the radicand. Show your work. (3 points)

Part B: Solve 4x^2 + 8x - 5 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used. (4 points)

Part C: Solve 2x^2 -12x + 5 = 0 by using a method different from the one you used in Part B. Show the steps of your work. (3 points)

Answer 2

This question needs knowledge of the quadratic formula.
Are you familiar with it?

Answer 3

A bit, yes @mathstudent55

Answer 4

The quadratic formula is:
\(\large x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
and solves the equation:
\(\large ax^2 + bx + c = 0\)

Answer 5

Look at your equation in part A.
\(\large 9x^2 - 16x + 60 = 0\)
What are a, b, and c for this equation?

Answer 6

A would be 9
b would be 16 or -16.. Im not sure
and c would be 60

Answer 7

Good.
a = 9; b = -16; c = 60
Notice b = -16.

Answer 8

Okay, so then since I have A b and c I can just plug in those values into the quadratic formula, right?

Answer 9

Yes, but don't do it yet.
The radicand is the expression of the quadratic formula that is inside the radical.
It is: \(b^2 - 4ac\)

Answer 10

The value of the radicand is called the "discriminant."
It tells you what the solutions will be like.

Answer 11

Let d = discriminant.

Answer 12

Okay so the it would be (-16)^2-4(9)(60)

Answer 13

If:
d > 0, there are two different real roots
d = 0, there are two equal real roots
d < =, there are two complex (imaginary) roots

Answer 14

Correct.
All they ask in part A is a description of the roots using the discriminant, so calculate the discriminant as you wrote it above.

Answer 15

(-16)^2-4(9)(60) = ?

Answer 16

(-16)^2= 256 so then it would become 256-4(9)(60)

Answer 17

good so far

Answer 18

4(9)(60) would become 2,160 right? making it 256-2160

Answer 19

still good

Answer 20

that equals to -1904

Answer 21

Great.
-1904 is obviously < 0, so what do you conclude about the nature of the roots?

Answer 22

that there are two complex (imaginary) roots

Answer 23

Excellent.
Part A is done.

Answer 24

Okay, Great ! Thank you c:

Answer 25

And part B?

Answer 26

If you want, you can try the quadratic formula or try factoring.

Answer 27

how would I factor?

Answer 28

We need to try to factor
\(\large 4x^2 + 8x - 5 = 0\)

Answer 29

This is a quadratic equation of the form:
\(\large ax^2 + bx + c = 0\)
where \(a \ne 1\)

Answer 30

We can use the ac method:
1. Multiply a and c together.

Answer 31

Okay, so then 4 multiplied with 5 is 20

Answer 32

Yes, but it really is 4 times -5, which is -20

Answer 33

2. Find two numbers that multiply to ac and add up to b
In our case, we need two numbers that multiply to -20 and add up to 8

Answer 34

okay so we will need a negative number because we need to multiply to -20

Answer 35

umm 10 and -2?

Answer 36

Correct. Also, the larger number must be positive bec they need to add up to 8, a positive number.

Answer 37

Great. 10 and -2 is correct.

Answer 38

3. Break up the middle term, bx, into a sum of two terms using these two numbers.
That means:
\(\large 4x^2 - 2x + 10x - 5 = 0\)

Answer 39

Notice that -2x + 10x = 8x, which is what we started with, so this new polynomial equals the original one.

Answer 40

4. Factor by grouping. Factor a common term from the first two terms, and factor a common term from the last two terms.

Answer 41

\(\large \color{blue}{4x^2 - 2x} + \color{red}{10x - 5} = 0\)

\(\large \color{blue}{2x}(2x - 1) + \color{red}{5}(2x - 1) = 0\)

Answer 42

okay so would I subtract 4 and 2

Answer 43

No, you can't subtract there since they are not like terms.
We factored out the common factor.

Answer 44

okay, so then what would I do with grouping them?

Answer 45

Now we have an expression with a common factor of 2x - 1
We factor out 2x - 1 to get our factored polynomial:
\(\large (2x - 1)(2x + 5) = 0\)

Answer 46

would we foil this?

Answer 47

Now we set each factor equal to zero and solve each equation for x.
\(2x - 1 = 0\) or \(2x + 5 = 0\)
2x = 1 or 2x = -5
x = 1/2 or x = -5/2

Answer 48

No, we don't FOIL it. We solve each equation to find the solution to the quadratic equation.

Answer 49

oh okay

Answer 50

sorry, im not that good a this

Answer 51

That is the solution to part B by factoring.
I have to go now, but for part C, since they want a different method, use the quadratic formula. Instead of only finding the discriminant, use the entire quadratic formula, plug in the values of a, b, and c from part C, and find x.

Answer 52

No problem.

Answer 53

If you try part C and post what you get, I'll look at it when I get back and will get back to you.

Answer 54

Sorry, gtg.

Answer 55

okay thank you for all your help

Answer 56

You're welcome.

Answer 57

x=17.1 or x=6.9 is what I got @mathstudent55

Answer 58

\(2x^2 -12x + 5 = 0\)

\(\large x = \dfrac{-(-12) \pm \sqrt{(-12)^2 - 4(2)(5)}}{2(2)}\)

\(\large x = \dfrac{12 \pm \sqrt{144 - 40}}{4}\)

\(\large x = \dfrac{12 \pm 2\sqrt{26}}{4}\)

\(\large x = \dfrac{6 \pm \sqrt{26}}{2}\)

\(x = 3 + \dfrac{\sqrt{26}}{2} \) or \(x = 3 - \dfrac{\sqrt{26}}{2} \)