Question

Part A:The area of a square is (4x2 − 12x + 9) square units. Determine the length of each side of the square by factoring the area expression completely. Show your work. (5 points)

Part B: The area of a rectangle is (16x2 − 9y2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)

Answer 1

@Mackenzie_Willa

Answer 2

@welshfella

Answer 3

I have the answer just not the steps!
Part A:(2x -3)^2
Part B:(4x+3y)(4x−3y)

Answer 4

@Nnesha

Answer 5

Do you know these?
\[a^2-b^2=(a-b)(a+b)
\\
(a+b)^2=a^2+2ab+b^2\]

Answer 6

Because the first shape is a square, the length and the width must be the same. So the area must be a perfect square. We need to find an expression that , when multiplied by itself, will give \(4x^2-12x+9\)Start by taking the square roots of the first and last coefficients. Can you do that?

Answer 7

Can you help me work it ut step by step plz

Answer 8

*out

Answer 9

What is the square root of the first term, \(4x^2\)?

Answer 10

2x^2?

Answer 11

Not quite. The square of 4 is 2, so you got that part right, but what is the square root of x^2? What can you multiply by itself to give x^2?

Answer 12

*square root of 4

Answer 13

I am confused

Answer 14

Do you understand that\[x \times x = x^2\]

Answer 15

yes

Answer 16

OK. So the square root of \(x^2\) is \(x\). OK with that?

Answer 17

oh okay.

Answer 18

All right. So the square root of \(4x^2\) is \(2x\). In other words\[2x \times 2x = 4x^2\]Make sense now?

Answer 19

Yes

Answer 20

OK. So now you need the square root of the last term, +9. What is it?

Answer 21

3

Answer 22

Great. With these two square roots, you have determined that the two factors must be\[4x^2-12x+9 = \left( 2x \text{ ? }3 \right)\left( 2x \text{ ? }3 \right)\]All that is left is to choose the correct sign, plus or minus. For that, look at the second term in the given trinomial. What sign does it have?

Answer 23

Minus.

Answer 24

Perfect. Then that is the sign you use in the factors. Therefore,\[4x^2-12x+9 = \left( 2x-3 \right)\left( 2x-3 \right)\]and you have your answer.

Answer 25

Keep in mind that this technique only works for factoring a perfect square trinomial.

Answer 26

Are these the answers. I have been working them out too:
Part A)
4x2 − 12x + 9
4x^2 -6x -6x + 9
2x( 2x - 3) -3(2x - 3)
(2x - 3)(2x-3)
length of each side is 2x-3

Part B)
16x^2 − 9y^2
(4x)^2 - (3y)^2
(4x - 3y) (4x+3y)
dimensions are 4x-3y, 4x+3y.

Answer 27

Yes, you are correct. Very well done!

Answer 28

Thank you!

Answer 29

You're welcome.

Answer 30

@ospreytriple can you help with 1 more?

Answer 31

An expression is shown below:

f(x) = –16x2 + 60x + 16

Part A: What are the x-intercepts of the graph of the f(x)? Show your work. (2 points)

Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)

Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)

Answer 32

Part A: To find the x-intercepts, you must solve the equation \(-16x^2+60x+16=0\). Which method would you use; factoring by completing the square, or the quadratic formula?

Answer 33

quadratic formula?

Answer 34

OK. Go ahead. What do you get?

Answer 35

Can you refresh me on the formula.

Answer 36

For the general quadratic, \(ax^2 + bx + c=0\), the x-intercepts are found using\[x=\frac{ -b \pm \sqrt{b^2-4ac} }{ 2a }\]

Answer 37

can you help me through this one

Answer 38

I am really confused.

Answer 39

Have you used the quadratic formula before?

Answer 40

Just learning it.

Answer 41

OK. I'll help you set it up.

Answer 42

Okay thank you! I do have 1 request can you say like part a and like part b and part c because i will get confused and don't know what to say.

Answer 43

This is for Part A. If you look at the given trinomial and compare it with the general quadratic, you'll see that \(a=-16\), \(b=60\) and \(c=16\). Putting these values into the quadratic formula, you have\[x=\frac{ -b \pm \sqrt{b^2-4ac} }{2a } = \frac{ -60 \pm \sqrt{60^2-4\left( -16 \right)\left( 16 \right)} }{ 2\left( -16 \right) }\]Can you complete these calculations?

Answer 44

is it -5?

Answer 45

One second. I'm working it out...

Answer 46

okay ;]

Answer 47

Wait is it...
x = 4 and x = -1/4.

Answer 48

Yayyyy! You got it. Those are the x-intercepts, \(x=4\) and \(x=-\frac{1}{4}\)

Answer 49

Yay :V.
can you help with part B and C too?

Answer 50

Part B: Vertex a maximum or minimum? What it means is, is the parabola opening upward or downward? If the parabola opens upward, then the vertex is a minimum. If the parabola opens downward, then the vertex is a maximum.
Do you know how to tell which direction the parabola opens?