Use graph of the function f(x)=x2 to find how the number of roots of the equation depends on the value of b.

a)x^2=x−b

If b < ANSWER, the equation has 2 roots.
If b = ANSWER, the equation has 1 root.
If b > ANSWER, the equation has no roots.

b) x^2=bx−1

If b is on the interval ( , ) ∪ ( , ), the equation has two roots.
If b equals to , , the equation has one root.
If b is on the interval ( , ), the equation has no roots.

  a)

  b)

Step-by-step explanation:

a) Writing the equation in standard form gives ...

  x^2 -x +b = 0

Then the discriminant* is ...

  d = (-1)^2 -4(1)(b) = 1-4b

Solving for b, we have ...

  b = (1 -d)/4

When d > 0, b < 1/4 and there will be 2 roots.

When d = 0, b = 1/4 and there will be 1 root.

When d < 0, b > 1/4 and there will be no real roots.

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b) We can work the second problem in similar fashion. The standard form of the given equation is ...

  x^2 -bx +1 = 0

The discriminant of this is ...

  d = (-b)^2 -4(1)(1) = b^2 -4

  d = (b +2)(b -2)

When d > 0, b ∈ (-∞, -2) ∪ (2, ∞) and there are 2 roots.

When d = 0, b ∈ {-2, 2}, and there is one root.

When d < 0, b ∈ (-2, 2), and there are no real roots.

* The discriminant of the equation ...

  ax^2 + bx + c = 0

is ...

  d = b^2 -4ac

When it is positive, there are two real roots; when it is zero, there is one real root; when it is negative, there are no real roots.

15

step-by-step explanation:

we are given the following expression and we are to solve it and find the product:

-13 × (-5)

solving the given expression in the standard order of solving a mathematical expression, beginning with the opening of brackets and then multiplying the terms and changing their signs.

-3 × -5 = 15 (answer is positive because when negative is multiplied with negative, it gives a positive answer)

d, x=8

step-by-step explanation:

simplifying

4x + -5 = 2x + 11

reorder the terms:

-5 + 4x = 2x + 11

reorder the terms:

-5 + 4x = 11 + 2x

solving

-5 + 4x = 11 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-5 + 4x + -2x = 11 + 2x + -2x

combine like terms: 4x + -2x = 2x

-5 + 2x = 11 + 2x + -2x

combine like terms: 2x + -2x = 0

-5 + 2x = 11 + 0

-5 + 2x = 11

add '5' to each side of the equation.

-5 + 5 + 2x = 11 + 5

combine like terms: -5 + 5 = 0

0 + 2x = 11 + 5

2x = 11 + 5

combine like terms: 11 + 5 = 16

2x = 16

divide each side by '2'.