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Let’s check out your problem:
A function
p
p
p
is defined as
p
(
x
)
=
(
x
−
a
)
(
x
−
15
)
(
x
−
20
)
+
15
p(x)=(x-a)(x-15)(x-20)+15
p
(
x
)
=
(
x
−
a
)
(
x
−
15
)
(
x
−
20
)
+
15
where
a
a
a
is a constant. Given that
p
(
7
)
=
15
p(7)=15
p
(
7
)
=
15
, what is the value of
a
a
a
?
\newline
□
\square
□
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Math Problems
Calculus
Find derivatives using implicit differentiation
Full solution
Q.
A function
p
p
p
is defined as
p
(
x
)
=
(
x
−
a
)
(
x
−
15
)
(
x
−
20
)
+
15
p(x)=(x-a)(x-15)(x-20)+15
p
(
x
)
=
(
x
−
a
)
(
x
−
15
)
(
x
−
20
)
+
15
where
a
a
a
is a constant. Given that
p
(
7
)
=
15
p(7)=15
p
(
7
)
=
15
, what is the value of
a
a
a
?
\newline
□
\square
□
Plug in
x
=
7
x=7
x
=
7
:
Plug
x
=
7
x=7
x
=
7
into the function to find
a
a
a
.
p
(
7
)
=
(
7
−
a
)
(
7
−
15
)
(
7
−
20
)
+
15
p(7)=(7-a)(7-15)(7-20)+15
p
(
7
)
=
(
7
−
a
)
(
7
−
15
)
(
7
−
20
)
+
15
Simplify the equation:
Simplify the equation.
p
(
7
)
=
(
7
−
a
)
(
−
8
)
(
−
13
)
+
15
p(7)=(7-a)(-8)(-13)+15
p
(
7
)
=
(
7
−
a
)
(
−
8
)
(
−
13
)
+
15
Set equation equal to
\newline
15
15
15
:
Since
\newline
p
(
7
)
=
15
p(7)=15
p
(
7
)
=
15
, set the equation equal to
\newline
15
15
15
and solve for
\newline
a
a
a
.
\newline
\newline
15
=
(
7
−
a
)
(
−
8
)
(
−
13
)
+
15
15=(7-a)(-8)(-13)+15
15
=
(
7
−
a
)
(
−
8
)
(
−
13
)
+
15
Isolate the variable term:
Subtract
15
15
15
from both sides to isolate the variable term.
\newline
0
=
(
7
−
a
)
(
−
8
)
(
−
13
)
0=(7-a)(-8)(-13)
0
=
(
7
−
a
)
(
−
8
)
(
−
13
)
Set
(
7
−
a
)
(7-a)
(
7
−
a
)
equal to
0
0
0
:
If the product of numbers is
0
0
0
, one of the numbers must be
0
0
0
. So, set
(
7
−
a
)
(7-a)
(
7
−
a
)
equal to
0
0
0
.
\newline
7
−
a
=
0
7-a=0
7
−
a
=
0
Solve for a:
Solve for
a
a
a
.
a
=
7
a=7
a
=
7
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