Question

Solve for x by completing the square of
f(x)=2x^2+7x+3

Help please i have part but the rest is so confusing!

Answer 1

sometimes it is just easier to use the quadratic formula, which is completing the square in one step

Answer 2

You need to solve \(2x^2 + 7x + 3 = 0\) by completing the square?

Answer 3

yes i do @mathstudent55

Answer 4

i already did that part @satellite73, i have to do this part too.

Answer 5

Ok. I'll tell you what the steps are. You do them and show me. I will go over them with you to make sure you're doing them correctly.

Answer 6

The first step is that you must have an equation of the form
\(x^2 + bx + c = 0\).
The coefficient of the \(x^2\) term must be 1, so divide both sides of the equation by 2.

Answer 7

no i was told by my teacher that completing the square was done differently @mathstudent55

Answer 8

This is how I do it.

\(2x^2+7x+3 = 0\)

Divide the entire equation by 2:

\(x^2+\dfrac{7}{2}x+\dfrac{3} {2} = 0\)

Answer 9

she told me i had to subtract 3 from both sides

Answer 10

Now subtract the constant term to the right side:

\(x^2+\dfrac{7}{2}x ~~~~~~~= -\dfrac{3} {2} \)

Answer 11

Ok. After you subtract the 3, then you divide both sides by 2?

Answer 12

so it will be x^2+3.5=-3/2?

Answer 13

@mathstudent55

Answer 14

Exactly.
That's what I have but with fractions.

Answer 15

Only you meant 3.5x, not just 3.5 on the left side.

Answer 16

right thought i typed the x. sorry, what do i do next? @mathstudent55

Answer 17

When you write the last equation, leave a space between the 3.5x and the = sign. That's where we will add the complete the square term.

\(x^2+3.5x~~~~~~~=-1.5\)

Answer 18

..okay so whats the next step? @mathstudent55

Answer 19

Let's go back to fractions.

\(x^2+\dfrac{7}{2}x ~~~~~~~= -\dfrac{3} {2}\)

The next step is the "complete the square step."

We take half of the coefficient of x, in this case 7/2, and we square it.
\(\left( \dfrac{7}{2} \right) = \dfrac{49}{4} \)

Answer 20

Then we add that to both sides.

\(x^2+\dfrac{7}{2}x +\color{red}{\dfrac{49}{4} }= -\dfrac{3} {2} +\color{red}{\dfrac{49}{4} }\)

Answer 21

okay so now what? @mathstudent55

Answer 22

Wait. Let's go back a few steps.
Let me write it again.

Answer 23

Let's go back to fractions.

\(x^2+\dfrac{7}{2}x ~~~~~ =−\dfrac{3}{2}\).

The next step is the "complete the square step."
We take half of the coefficient of x, in this case \(\dfrac{7}{2}\).

Answer 24

Okay now I'm super confused @mathstudent55

Answer 25

\(\dfrac{7}{2}\) is the coefficient of x.
We take half of it: \(\dfrac{7}{4} \)
Now we square that: \(\dfrac{49}{16} \)

Answer 26

how is half of 7/2 7/4? @mathstudent55

Answer 27

Ok. I am sorry. I wrote above what to do but did it incorrectly.
We need to take the coefficient of the x term. It's 7/2.

Answer 28

The we need to take half of it.
Half of 7/2 is 7/4.

Answer 29

okay...? @mathstudent55 so instead of 49/4 its 49/16?

Answer 30

This is why: \(\dfrac{7}{2} \div 2 = \dfrac{7}{2} \div \dfrac{2}{1} = \dfrac{7}{2} \times \dfrac{1}{2} = \dfrac{7}{4}\)

Answer 31

Correct.

Answer 32

Now we square 7/4 to get 49/16.

Answer 33

That's what we add to both sides.

Answer 34

We had this.

\(x^2+\dfrac{7}{2}x ~~~~~ =−\dfrac{3}{2}\)

Now we add 49/16 to both sides:

\(x^2+\dfrac{7}{2}x + \dfrac{49}{16} =−\dfrac{3}{2} + \dfrac{49}{16}\)

Answer 35

okay

Answer 36

Since this whole procedure is completing the square on the left side, the left side now is the square of a binomial, so we write the left side as \((x + \dfrac{7}{4})^2 \).

Answer 37

\(\left(x+\dfrac{7}{4} \right)^2 =−\dfrac{3}{2} + \dfrac{49}{16} \)

Now we add the fractions on the right side. We need to use the LCD of 16.

Answer 38

now i'm confused

Answer 39

and i have a serious time limit i have to go teach bible school in 15 minutes

Answer 40

@mathstudent55

Answer 41

We're almost done.
Ask your question. Where are you confused?

Answer 42

how does the left side go from x^2+7/2x to a 7/4?

Answer 43

where did the 49/16 go?

Answer 44

@mathstudent55

Answer 45

The entire idea behind the "complete the square" method is to turn the left side into a perfect square trinomial.

Answer 46

so do the 7/2x +49/16 =7/4? @mathstudent55

Answer 47

Remember the pattern: \((a + b)^2 = a^2 + 2ab + b^2\)
We are doing it in reverse.

Answer 48

but am i correct?@math&ing001

Answer 49

We took \(a^2 + 2ab\).
Then we figure out what we needed to add to get the \(b^2\) part.
We added it.
Then we went from
\(a^2 + 2ab + b^2 \) to \((a + b)^2\).

Answer 50

okay no what now? @mathstudent55

Answer 51

We applied the pattern to do this to the left side.
\(x^2 + \dfrac{7}{2}x + \dfrac{49}{16} = \left(x+\dfrac{7}{4} \right)^2 \)

Answer 52

yes i know, how do i solve for x? from here? @mathstudent55

Answer 53

Now we add the fractions on the right side suing a common denominator.

\(\left(x+\dfrac{7}{4} \right)^2 =−\dfrac{3}{2} \times \dfrac{8}{8} + \dfrac{49}{16} \)

\(\left(x+\dfrac{7}{4} \right)^2 =−\dfrac{24}{16} + \dfrac{49}{16} \)

\(\left(x+\dfrac{7}{4} \right)^2 =\dfrac{25}{16} \)

Now there is a principle that if x^2 = k (k is a number), then
\(x = \sqrt{k}\) or \(x = -\sqrt{k} \). We apply the principle to our situation:

\(x+\dfrac{7}{4} =\sqrt{\dfrac{25}{16}} \) or \(x+\dfrac{7}{4} =-\sqrt{\dfrac{25}{16}} \)

\(x+\dfrac{7}{4} =\dfrac{5}{4} \) or \(x+\dfrac{7}{4} =-\dfrac{5}{4} \)

\(x =-\dfrac{2}{4} \) or \(x =-\dfrac{12}{4} \)

\(x =-\dfrac{1}{2} \) or \(x =-3 \)

Answer 54

THANK YOU SO MUCH! @mathstudent55

Answer 55

You're welcome.