Question

Using the completing-the-square method, rewrite f(x) = x2 + 4x − 1 in vertex form.

Answer 1

nah do it the easy way

Answer 2

@satellite73 do you know how to do it?

Answer 3

there is a much easier way that "completing the square" not that it is not useful to know

Answer 4

the choices are

f(x) = (x + 2)2 + 1
f(x) = (x + 2)2
f(x) = (x + 2)2 + 4
f(x) = (x + 2)2 − 5

Answer 5

yes i know how to do it
the first coordinate of the vertex of \(y=ax^2+bx+c\) is always \(-\frac{b}{2a}\)
in your case \(a=1,b=4\) and \(-\frac{b}{2a}=-2\)

Answer 6

so you know it will be \[f(x)=(x+2)^2+\text{something}\]

Answer 7

to find the "something" replace \(x\) by \(-2\) and compute \[y=(-2)^2+4\times (-2)+1\]

Answer 8

look here im going to give you an example

Example: y = x^2 + 4x - 6

You have to complete the square to convert it into vertex form.

y = x² + 4x - 6

Group.
y = (x² + 4x) - 6

Factor
y = (x² + 4x) - 6

Add placeholders.
y = (x² + 4x + ___) - 6 - 1(___)

Notice that the second blank is multiplied by -1 to account for what you had to add to complete the square.

Take the coefficient of the x term: 4
Divide it by 2: 4 / 2 = 2
Square it: (2)² = 4

Add 4 to both blanks.
y = (x² + 4x + 4) - 6 - 1(4)

x² + 4x + 4 is the expanded form of a perfect square binomial.

Remember that (a + b)² = a² + 2ab + b². Apply this to what you have.
y = (x² + 4x + 4) - 6 - 1(4)
y = (x + 2)² - 6 - 1(4)

Simplify the rest.
y = (x + 2)² - 6 - 4
y = (x + 2)² - 10

Remember that the vertex form is: y = a(x - h)² + k

CHECK:
y = (x + 2)² - 10
y = [(x)² + 2(x)(2) + (2)²] - 10
y = (x² + 4x + 4) - 10
y = (x²) + 1(4x) + 1(4) - 10
y = x² + 4x + 4 - 10
y = x² + 4x - 6
TRUE

ANSWER: y = (x + 2)² - 10 is the vertex form.

Given: y = (x + 2)² - 10
Means: h = -2
Means: k = -10
Means: a = 1

ANSWER: The vertex is at (-2, -10).

Since the equation is a function of x and a is positive, the parabola opens upwards.

Parabolas that open upwards or downwards have an axis of symmetry that is the same as the h coordinate of the vertex.

ANSWER: The axis of symmetry is at x = -2.

Remember that y-intercepts have x = 0. Find the value of y when x = 0 by substitution x with 0 in the original equation.

y = x² + 4x - 6
y = 0² + 4(0) - 6
y = 0 + 0 - 6
y = -6

ANSWER: The y-intercept is at (0, -6).

Answer 9

that is the number out at the end

Answer 10

is that helpful 0_o

Answer 11

wait so the answer to my question would be f(x) = (x + 2)^2 + 1 ?

Answer 12

@Taylor0402 @satellite73

Answer 13

hold on let me check

Answer 14

no

Answer 15

\[(-2)^2+4\times (-2)+1\] is not \(1\)

Answer 16

I got -7 for the answer to that @satellite73

Answer 17

really?

Answer 18

\[(-2)^2+4\times (-2)+1\neq -7\]

Answer 19

is it -3? i dont know how im messing this up @satellite73

Answer 20

\[4-8+1=-4+1=-3\]

Answer 21

oooh hold on, typeo!!

Answer 22

it is \[x^2+4x-1\] so \[4-8-1=-5\]

Answer 23

i put plus one, it should have been minus one, my fault

Answer 24

oooh okay! so it's f(x) = (x + 2)^2 − 5

Answer 25

can you help me with another one? @satellite73

Answer 26

sure

Answer 27

Solve x2 − 12x + 5 = 0 using the completing-the-square method. @satellite73

Answer 28

ok this time we really will complete the square, because it is easiest done that way

Answer 29

first subtract 5 from both sides to get \[x^2-12x=-5\]
now, what is half of 12?

Answer 30

6 @satellite73

Answer 31

right
and \(6^2=?\)

Answer 32

36

Answer 33

exactly
so we go from \[x^2-12x=-5\] to \[(x-6)^2=-5+36\] or \[(x-6)^2=31\] that is called "completing the square"

Answer 34

now we are ready to solve it \[(x-6)^2=31\\
x-6=\pm\sqrt{31}\\
x=6\pm\sqrt{31}\]

Answer 35

thank you so much!!!! can you help me with one more? @satellite73

Answer 36

ok but this one will cost you $7

Answer 37

just kidding

Answer 38

thank you!!!! okay its
Rewrite f(x) = –2(x − 3)2 + 2 from vertex form to standard form.
i dont know the difference b/w vertex and standard

Answer 39

it just means multiply out and combine like terms to make it look like \[f(x)=ax^2+bx+c\]

Answer 40

i.e. write out \[-2(x-3)(x-3)+2\] multiply, distribute the \(-2\) etc

Answer 41

@satellite73 okay so, f(x) = 4x^2 - 24x + 38 ?

Answer 42

that cannot be right

Answer 43

don't square the \(-2\)

Answer 44

square the \(x-3\) i.e. compute \((x-3)(x-3)\) then multipiy the result by \(-2\)

Answer 45

f(x) = -2x + 12x - something? @satellite73

Answer 46

f(x) = –2x2 + 12x − 16 !!! right?

Answer 47

sorry -2x^2 + 12x - 16