A binary operation ∆ defined on the R of real numbers by a∆b=A square-2ab+bsquare where a,bER find (a)R such that 2∆(-5)=√8. (b) (x+1)∆x out of x, x is not equal to 0.

Question

A binary operation ∆ defined on the R of real numbers by a∆b=A square-2ab+bsquare where a,bER find (a)R such that 2∆(-5)=√8. (b) (x+1)∆x out of x, x is not equal to 0.

Answer 1

Solve for this

Answer 2

a∆b = a^2-2ab+b^2 = (a-b)^2

(a) Huh? 2∆(-5) = (2+5)^2 = 49
It cannot be √8
What does R have to do with it?

(b) (x+1)∆x = (x+1-x)^2 = 1
what does "out of x" mean?

I think you have garbled the question.

Answer 3

To solve part (a), we need to find the value of a in the real numbers (R) such that 2∆(-5) = √8.

Let's substitute the values into the expression:
2∆(-5) = (√8)^2 - 2(-5)(√8) + (-5)^2

Simplifying further:
2∆(-5) = 8 - 2 * -5 * √8 + 25
2∆(-5) = 8 + 10√8 + 25
2∆(-5) = 33 + 10√8

Now, since we want this expression to equal √8, we can set it equal to √8:
33 + 10√8 = √8

Subtracting √8 from both sides:
33 = √8 - 10√8

Combining like terms:
33 = -9√8

Dividing both sides by -9 (since √8 cannot be equal to 0):
√8 = -33/-9

Simplifying the right side:
√8 = 33/9
√8 = 11/3

However, this is a contradiction since √8 is irrational and cannot be equal to 11/3, which is rational. Therefore, there is no value in R that satisfies the equation 2∆(-5) = √8.

Now, let's move on to part (b).

To find (x+1)∆x, we substitute the values into the expression:

(x+1)∆x = (x+1)^2 - 2(x)(x+1) + (x)^2

Simplifying further:
(x+1)∆x = (x^2 + 2x + 1) - 2x(x+1) + x^2

Expanding and simplifying the expression:
(x+1)∆x = x^2 + 2x + 1 - 2x^2 - 2x + x^2
(x+1)∆x = -x^2 + 1

So, (x+1)∆x simplifies to -x^2 + 1, where x is not equal to 0.

Answer 4

To find the values of (a) and (b) in the given binary operation, let's go step by step:

(a) We need to find the value of 'a' in the real numbers (R) that satisfies the equation 2∆(-5) = √8.

First, let's substitute the given values into the expression:
2∆(-5) = (√8)^2 - 2(2)(-5) + (-5)^2

Simplifying further:
2∆(-5) = 8 - 4(-5) + 25
2∆(-5) = 8 + 20 + 25
2∆(-5) = 53

Now, let's solve for 'a' using the definition of the binary operation:
a∆b = a^2 - 2ab + b^2

Substituting the values we found so far:
a^2 - 2a(-5) + (-5)^2 = 53
a^2 + 10a + 25 = 53
a^2 + 10a + 25 - 53 = 0
a^2 + 10a - 28 = 0

We can solve this quadratic equation using factoring or the quadratic formula. By factoring, we have:
(a + 14)(a - 2) = 0

Setting each factor equal to zero gives two possible values for 'a':
a + 14 = 0 -> a = -14
a - 2 = 0 -> a = 2

So, in this case, there are two possible values for 'a' in the real numbers (R) that satisfy the equation 2∆(-5) = √8: a = -14 and a = 2.

(b) Now, let's find the expression (x + 1)∆x, where x is not equal to 0.

Using the definition of the binary operation:
(x + 1)∆x = (x + 1)^2 - 2(x + 1)x + x^2

Expanding and simplifying:
(x + 1)∆x = (x^2 + 2x + 1) - 2(x^2 + x) + x^2
(x + 1)∆x = x^2 + 2x + 1 - 2x^2 - 2x + x^2
(x + 1)∆x = -x^2

So, the expression (x + 1)∆x simplifies to -x^2 when x is not equal to 0.