Question

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.

Answer 1

Here's what I get

Explanation:

a) x² = x - b

I plotted the graphs of y = x² - x + b with different values of b and found:

`\begin{cases}\text{0 real roots} & \quad \text{if } b < 0.25 \\\text{1 real root}& \quad \text{if } b = 0.25\\\text{2 real roots}& \quad \text{if } b > 0.25\\\end{cases}`

You can see three specific cases in Fig.1.

b) x² = bx - 1

I plotted the graphs of y = x² - bx + 1 with different values of b and found:

`\begin{cases}\text{2 real roots} & \quad \text{if } b \text{ is on the interval } (-\infty, -2) \cup (2,\infty) \\\text{1 real root}& \quad \text{if } b = -2 \text{ or } 2\\\text{0 real roots}& \quad \text{if } b \text{ is on the interval (-2,2)}\\\end{cases}`

You can see five specific cases in Fig. 2.