Solve for the dimensions of the area model below and then write an equation showing how the area as a sum is equivalent to the a
1 answer:
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To solve this, you can do it the easy way or hard way
easy way is to know some stuff about parabolas and zeroes
hard way is to subsitute for x and f(x) and solve for a,b, and c with a system of 3 variables (harder)
I'm going to do the easy way
nice we are given some zeroes
when f(x)=0, x=-2
since it is quadratic, it is 2nd degree and we can write all 2nd degree functions in the form f(x)=a(x-b)(x-c)
where a is some random constant and b and c are the x intercepts
we are given that when f(x)=0, x=-2, that means one x intercept is at -2
so lets say b=-2
f(x)=a(x-(-2))(x-c)
f(x)=a(x+2)(x-c)
great, now to find the other values
when x=0, y=-6
subsitute 0 for x and see what happens
-6=a(0+2)(0-c)
-6=a(2)(-c)
-6=-2ac
divide by -2 both sides
3=ac
cool
we revisit our equation
f(x)=a(x+2)(x-c)
now if we solve for a, we can do
3=ac, a=3/c, subsitute 3/c for a
f(x)=(3/c)(x+2)(x-c)
subsitute (4,78)
78=(3/c)(4+2)(4-c)
78=(3/c)(6)(4-c)
divide both sides by 6 for ease
13=(3/c)(4-c)
13=12/c-3
add 3 to both sides
16=12/c
multiply both sides by c
16c=12
divide both sides by 16
c=12/16
c=0.75
find a
a=3/c
a=3/0.75
a=4
if might be best to leave in fraction form
a=4
c=0.75=3/4
the equation is
in standard form as in ax^2+bx+c form
easy way is to know some stuff about parabolas and zeroes
hard way is to subsitute for x and f(x) and solve for a,b, and c with a system of 3 variables (harder)
I'm going to do the easy way
nice we are given some zeroes
when f(x)=0, x=-2
since it is quadratic, it is 2nd degree and we can write all 2nd degree functions in the form f(x)=a(x-b)(x-c)
where a is some random constant and b and c are the x intercepts
we are given that when f(x)=0, x=-2, that means one x intercept is at -2
so lets say b=-2
f(x)=a(x-(-2))(x-c)
f(x)=a(x+2)(x-c)
great, now to find the other values
when x=0, y=-6
subsitute 0 for x and see what happens
-6=a(0+2)(0-c)
-6=a(2)(-c)
-6=-2ac
divide by -2 both sides
3=ac
cool
we revisit our equation
f(x)=a(x+2)(x-c)
now if we solve for a, we can do
3=ac, a=3/c, subsitute 3/c for a
f(x)=(3/c)(x+2)(x-c)
subsitute (4,78)
78=(3/c)(4+2)(4-c)
78=(3/c)(6)(4-c)
divide both sides by 6 for ease
13=(3/c)(4-c)
13=12/c-3
add 3 to both sides
16=12/c
multiply both sides by c
16c=12
divide both sides by 16
c=12/16
c=0.75
find a
a=3/c
a=3/0.75
a=4
if might be best to leave in fraction form
a=4
c=0.75=3/4
the equation is
in standard form as in ax^2+bx+c form
Answer:
c < 6/19
Step-by-step explanation: