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.
if b>2 or b<-2 has 2 roots,
if -2 or 2 1 root,
and if -2<b<2 it has no roots
To determine the number of roots of the equation based on the graph of the given function, we need to analyze the intersection points between the graph of f(x) = x^2 and the equation in question.
a) For the equation x^2 = x - b:
To find the roots of this equation, we need to set the equation equal to zero:
x^2 - x + b = 0
The discriminant of the quadratic equation is Δ = b^2 - 4ac, where a = 1, b = -1, and c = b.
If the discriminant is positive (Δ > 0), the equation has two distinct real roots.
If the discriminant is zero (Δ = 0), the equation has one real root.
If the discriminant is negative (Δ < 0), the equation has no real roots.
Now, let's consider the graph of f(x) = x^2:
The graph of f(x) = x^2 is a parabola that opens upwards. It intersects the x-axis at the point (0,0) and is symmetric about the y-axis.
By comparing the equation x^2 = x - b with the graph of f(x) = x^2:
- If b < 0, the graph of f(x) = x^2 is shifted vertically upward by |b| units. In this case, the parabola will intersect the x-axis at two distinct real points, so the equation will have 2 roots.
- If b = 0, the graph of f(x) = x^2 remains the same, and there will be one intersection point, hence the equation will have 1 root.
- If b > 0, the graph of f(x) = x^2 is shifted vertically downward by b units. In this case, the parabola will not intersect the x-axis, so the equation will have no real roots.
Therefore, if b < 0, the equation x^2 = x - b has 2 roots.
If b = 0, the equation x^2 = x - b has 1 root.
If b > 0, the equation x^2 = x - b has no roots.
b) For the equation x^2 = bx - 1:
To find the roots of this equation, we need to set the equation equal to zero:
x^2 - bx + 1 = 0
Again, we calculate the discriminant Δ = b^2 - 4ac, where a = 1, b = -b, and c = 1.
Analyzing the graph of f(x) = x^2:
- The parabola opens upwards.
- The vertex of the parabola is at the point (0,0).
- The parabola intersects the x-axis at two points symmetrically on either side of the y-axis.
By comparing the equation x^2 = bx - 1 with the graph of f(x) = x^2:
- If b is in the interval (0, 4), the parabola intersects the x-axis at two distinct real points. Therefore, the equation has two roots.
- If b equals 0 or 4, the parabola intersects the x-axis at one point, so the equation has 1 root.
- If b is less than 0 or greater than 4, the parabola does not intersect the x-axis, hence the equation has no real roots.
Therefore, if b is in the interval (0, 4), the equation x^2 = bx - 1 has 2 roots.
If b equals 0 or 4, the equation x^2 = bx - 1 has 1 root.
If b is less than 0 or greater than 4, the equation x^2 = bx - 1 has no roots.
Check the discriminant, which is equal to
(a) 1 - 4b
(b) b^2-4
Now recall what the discriminant means ...