Question

Find the values of a and b that turn equation into identity

Answer 1

(2x+5) / (x-8)(2x+1) = (a/x-8) + (b/2x+1)

Let's find the values of a and b by equating the numerators:

2x + 5 = a(2x+1) + b(x-8)

2x + 5 = 2ax + a + bx - 8b

now we will get X coefficient

2 = 2a + b

and now we will get the constant term

5 = a - 8b

so

eq1:.

2a + b = 2

and
eq2:.

a - 8b = 5

from the first equation

2 = 2a + b

2 - b = 2a

-b = 2a - 2

b = -2a + 2

and now we will substitute equation 2

a - 8b = 5

a - 8 ( -2a + 2) = 5

a + 16a - 16 = 5

a + 16a = 5 + 16

17a = 21

a = ²¹/₁₇ = 1.24

1.24 - 8b = 5

-8b = 5 - 1.24

-8b = 3.76

b = -0.47

Answer 2

`\sf a = (21)/(17)`

`\sf b = -(8)/(17)`

Explanation:

To find the values of
`\sf a` and
`\sf b` such that the given expression becomes an identity, we can use partial fraction decomposition. The expression in question is:

`\sf (2x + 5)/((x - 8)(2x + 1))`

The denominator can be factored as:

`\sf (x - 8)(2x + 1)`.

Now, we express the given fraction as the sum of two fractions with unknown values
`\sf a` and
`\sf b`:

`\sf (2x + 5)/((x - 8)(2x + 1)) = (a)/(x - 8) + (b)/(2x + 1)`

To find
`\sf a` and
`\sf b`, we can clear the denominators by multiplying both sides of the equation by the common denominator
`\sf (x - 8)(2x + 1)`. This gives:

`\sf 2x + 5 = a(2x + 1) + b(x - 8)`

Now, we can solve for
`\sf a` and
`\sf b` by comparing coefficients. Let's expand the right side:

`\sf 2x + 5 = 2ax + a + bx - 8b`

Now, group like terms:

`\sf 2x + 5 = (2a + b)x + (a - 8b)`

By comparing coefficients, we can write a system of equations:

`\sf 2a + b = 2`

`\sf a - 8b = 5`

Solving this system will give us the values of
`\sf a` and
`\sf b` that turn the equation into an identity. Let's solve the system:

`\sf 2a + b = 2`

`\sf a - 8b = 5`

Multiplying the second equation by 2 to match coefficients:

`\sf 2a + b = 2`

`\sf 2a - 16b = 10`

Subtracting the first equation from the second:

`\sf 2a - 16b -2a-b = 10-2`

`\sf -17b = 8`

`\sf b = -(8)/(17)`

Now, substitute the value of
`\sf b` back into one of the original equations. Using the first equation:

`\sf 2a - (8)/(17) = 2`

Multiply both sides by 17 to get rid of the fraction:

`\sf 2a\cdot 17 - (8)/(17) \cdot 17= 2\cdot 17`

`\sf 34a - 8 = 34`

`\sf 34a = 42`

`\sf a = (21)/(17)`

So, the values of
`\sf a` and
`\sf b` that turn the given expression into an identity are:

`\sf a = (21)/(17)`

`\sf b = -(8)/(17)`