Given that the quadratic equation x^2-2x-m+1=0 has 2 positive real number roots, a and b, determine the range of the values of m.
b^2-4ac is positive for real roots
4 - 4(1)(1-m) is positive
4 - 4 + 4 m is positive
or 4 m is positive
so m is positive
m>0 for sure
now
2 +/- sqrt (b^2-4ac) is positive for positive root
2 +/- sqrt(4-4(1-m) ) is positive
2 +/- sqrt(m) is positive
m < 4
so
0 <m< 4
To find the range of values of m for which the quadratic equation has 2 positive real number roots, we need to consider the discriminant of the equation.
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
For the given quadratic equation x^2 - 2x - m + 1 = 0, we can compare it with the general form ax^2 + bx + c = 0 and get:
a = 1
b = -2
c = -m + 1
Now, we can calculate the discriminant:
D = (-2)^2 - 4(1)(-m + 1)
= 4 + 4m - 4
= 4m
Since the equation has 2 positive real number roots, the discriminant (4m) must be greater than 0. Therefore,
4m > 0
Dividing both sides by 4, we get:
m > 0
Thus, the range of values for m is m > 0.
To determine the range of values of m, we need to consider the discriminant of the quadratic equation. The discriminant, denoted as Δ, is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In the given quadratic equation x^2 - 2x - m + 1 = 0, we have a = 1, b = -2, and c = -m + 1. Substituting these values into the discriminant formula, we have:
Δ = (-2)^2 - 4(1)(-m + 1)
= 4 + 4m - 4
= 4m
For there to be two positive real number roots (a and b), the discriminant Δ must be positive. Therefore, we have:
4m > 0
Dividing both sides of the inequality by 4 (since 4 is positive), we get:
m > 0/4
m > 0
Therefore, the range of values for m is m > 0. In other words, any positive real number will satisfy the given conditions.