Use graph of the function f(x)=x2 to find how the number of roots of the equation depends on the value of 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.
To understand how the number of roots of the equation x^2 = bx - 1 depends on the value of b, we can use the graph of the function f(x) = x^2.
1. Start by plotting the graph of the function f(x) = x^2. This is a U-shaped curve that opens upward, with the vertex at the origin (0,0).
2. Next, let's analyze the equation x^2 = bx - 1 in terms of the graph. This equation represents the intersection points between the graph of f(x) = x^2 and the line y = bx - 1.
3. Consider the case when b is on the interval (−∞, -2) ∪ (2, +∞). In this range, the line y = bx - 1 will not intersect the graph of f(x) = x^2. Therefore, there are no intersection points and, hence, no roots to the equation x^2 = bx - 1.
4. Now, let's look at the case when b equals -2 or 2. For b = -2, the line y = bx - 1 will be tangent to the graph of f(x) = x^2 at exactly one point. This means there is only one intersection point, which corresponds to one root for the equation x^2 = bx - 1. Similarly, when b = 2, we have the same scenario with one intersection point and one root.
5. Finally, consider the case when b is on the interval (-2, 2). In this range, the line y = bx - 1 intersects the graph of f(x) = x^2 at two distinct points. Therefore, there are two intersection points which correspond to two roots for the equation x^2 = bx - 1.
In summary:
- If b is on the interval (−∞, -2) ∪ (2, +∞), the equation x^2 = bx - 1 has no roots.
- If b equals -2 or 2, the equation x^2 = bx - 1 has one root.
- If b is on the interval (-2, 2), the equation x^2 = bx - 1 has two roots.
By using the graph of the function f(x) = x^2, we can visually analyze and understand how the equation's roots vary with different values of b.
remember the discriminant. It tells you about the roots. In this case, you have
x^2 - bx + 1 = 0
so the discriminant is b^2 - 4