Solved on Jan 18, 2024

Find the value of $a$ when $f(x)=2x^3-9x^2+7x+a=0$ with $x=2$ .

STEP 1

Assumptions
1. The function given is $f(x) = 2x^3 - 9x^2 + 7x + a$ .
2. We need to find the value of $a$ when $x = 2$ and $f(x) = 0$ .

STEP 2

We will substitute $x = 2$ into the function $f(x)$ and set the function equal to zero.
$f(2) = 2(2)^3 - 9(2)^2 + 7(2) + a$

STEP 3

Calculate the value of $2^3$ which is $2 \times 2 \times 2$ .
$2^3 = 8$

STEP 4

Calculate the value of $9(2)^2$ which is $9 \times 2 \times 2$ .
$9(2)^2 = 9 \times 4 = 36$

STEP 5

Calculate the value of $7(2)$ which is $7 \times 2$ .
$7(2) = 14$

STEP 6

Substitute the calculated values back into the function.
$f(2) = 2(8) - 36 + 14 + a$

STEP 7

Multiply $2$ by $8$ .
$2(8) = 16$

STEP 8

Now, combine all the constant terms together.
$f(2) = 16 - 36 + 14 + a$

STEP 9

Simplify the constant terms by performing the arithmetic operations.
$f(2) = -20 + 14 + a$
$f(2) = -6 + a$

STEP 10

Since we know that $f(2) = 0$ , we can set the expression equal to zero and solve for $a$ .
$0 = -6 + a$

STEP 11

Add $6$ to both sides of the equation to isolate $a$ .
$0 + 6 = -6 + a + 6$

STEP 12

Simplify the equation to find the value of $a$ .
$a = 6$
The value of $a$ when $f(x) = 0$ and $x = 2$ is $a = 6$ .

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Find the value of $a$ when $f(x)=2x^3-9x^2+7x+a=0$ with $x=2$ .

STEP 1

Assumptions
1. The function given is $f(x) = 2x^3 - 9x^2 + 7x + a$ .
2. We need to find the value of $a$ when $x = 2$ and $f(x) = 0$ .

STEP 2

We will substitute $x = 2$ into the function $f(x)$ and set the function equal to zero.
$f(2) = 2(2)^3 - 9(2)^2 + 7(2) + a$

STEP 3

Calculate the value of $2^3$ which is $2 \times 2 \times 2$ .
$2^3 = 8$

STEP 4

Calculate the value of $9(2)^2$ which is $9 \times 2 \times 2$ .
$9(2)^2 = 9 \times 4 = 36$

STEP 5

Calculate the value of $7(2)$ which is $7 \times 2$ .
$7(2) = 14$

STEP 6

Substitute the calculated values back into the function.
$f(2) = 2(8) - 36 + 14 + a$

STEP 7

Multiply $2$ by $8$ .
$2(8) = 16$

STEP 8

Now, combine all the constant terms together.
$f(2) = 16 - 36 + 14 + a$

STEP 9

Simplify the constant terms by performing the arithmetic operations.
$f(2) = -20 + 14 + a$
$f(2) = -6 + a$

STEP 10

Since we know that $f(2) = 0$ , we can set the expression equal to zero and solve for $a$ .
$0 = -6 + a$

STEP 11

Add $6$ to both sides of the equation to isolate $a$ .
$0 + 6 = -6 + a + 6$

STEP 12

Simplify the equation to find the value of $a$ .
$a = 6$
The value of $a$ when $f(x) = 0$ and $x = 2$ is $a = 6$ .

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Find the value of aaa when f(x)=2x3−9x2+7x+a=0f(x)=2x^3-9x^2+7x+a=0f(x)=2x3−9x2+7x+a=0 with x=2x=2x=2.

STEP 1

Assumptions1. The function given is f(x)=2x3−9x2+7x+a f(x) = 2x^3 - 9x^2 + 7x + a f(x)=2x3−9x2+7x+a.2. We need to find the value of a a a when x=2 x = 2 x=2 and f(x)=0 f(x) = 0 f(x)=0.

STEP 2

We will substitute x=2 x = 2 x=2 into the function f(x) f(x) f(x) and set the function equal to zero.f(2)=2(2)3−9(2)2+7(2)+a f(2) = 2(2)^3 - 9(2)^2 + 7(2) + a f(2)=2(2)3−9(2)2+7(2)+a

STEP 3

Calculate the value of 23 2^3 23 which is 2×2×2 2 \times 2 \times 2 2×2×2.23=8 2^3 = 8 23=8

STEP 4

Calculate the value of 9(2)2 9(2)^2 9(2)2 which is 9×2×2 9 \times 2 \times 2 9×2×2.9(2)2=9×4=36 9(2)^2 = 9 \times 4 = 36 9(2)2=9×4=36

STEP 5

Calculate the value of 7(2) 7(2) 7(2) which is 7×2 7 \times 2 7×2.7(2)=14 7(2) = 14 7(2)=14

STEP 6

Substitute the calculated values back into the function.f(2)=2(8)−36+14+a f(2) = 2(8) - 36 + 14 + a f(2)=2(8)−36+14+a

STEP 7

Multiply 2 2 2 by 8 8 8.2(8)=16 2(8) = 16 2(8)=16

STEP 8

Now, combine all the constant terms together.f(2)=16−36+14+a f(2) = 16 - 36 + 14 + a f(2)=16−36+14+a

STEP 9

Simplify the constant terms by performing the arithmetic operations.f(2)=−20+14+a f(2) = -20 + 14 + a f(2)=−20+14+af(2)=−6+a f(2) = -6 + a f(2)=−6+a

STEP 10

Since we know that f(2)=0 f(2) = 0 f(2)=0, we can set the expression equal to zero and solve for a a a.0=−6+a 0 = -6 + a 0=−6+a

STEP 11

Add 6 6 6 to both sides of the equation to isolate a a a.0+6=−6+a+6 0 + 6 = -6 + a + 6 0+6=−6+a+6

STEP 12

Simplify the equation to find the value of a a a.a=6 a = 6 a=6The value of a a a when f(x)=0 f(x) = 0 f(x)=0 and x=2 x = 2 x=2 is a=6 a = 6 a=6.

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Find the value of $a$ when $f(x)=2x^3-9x^2+7x+a=0$ with $x=2$ .

Assumptions
1. The function given is $f(x) = 2x^3 - 9x^2 + 7x + a$ .
2. We need to find the value of $a$ when $x = 2$ and $f(x) = 0$ .

We will substitute $x = 2$ into the function $f(x)$ and set the function equal to zero.
$f(2) = 2(2)^3 - 9(2)^2 + 7(2) + a$

Calculate the value of $2^3$ which is $2 \times 2 \times 2$ .
$2^3 = 8$

Calculate the value of $9(2)^2$ which is $9 \times 2 \times 2$ .
$9(2)^2 = 9 \times 4 = 36$

Calculate the value of $7(2)$ which is $7 \times 2$ .
$7(2) = 14$

Substitute the calculated values back into the function.
$f(2) = 2(8) - 36 + 14 + a$

Multiply $2$ by $8$ .
$2(8) = 16$

Now, combine all the constant terms together.
$f(2) = 16 - 36 + 14 + a$

Simplify the constant terms by performing the arithmetic operations.
$f(2) = -20 + 14 + a$
$f(2) = -6 + a$

Since we know that $f(2) = 0$ , we can set the expression equal to zero and solve for $a$ .
$0 = -6 + a$

Add $6$ to both sides of the equation to isolate $a$ .
$0 + 6 = -6 + a + 6$

Simplify the equation to find the value of $a$ .
$a = 6$
The value of $a$ when $f(x) = 0$ and $x = 2$ is $a = 6$ .