whate must the value of k
2Xsq+(k-3)x+k-5=0 has one real root ? Two real root ?no real root?
Zed
To determine the number of real roots of the quadratic equation, we need to consider the discriminant (denoted as Δ) of the equation. The discriminant is calculated using 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 equation 2x^2 + (k-3)x + (k-5) = 0, we can identify that:
a = 2 (coefficient of x^2),
b = (k-3) (coefficient of x),
c = (k-5) (constant term).
To determine the number of real roots, we need to evaluate the discriminant, as follows:
1. If Δ > 0, the equation has two distinct real roots.
2. If Δ = 0, the equation has one real root.
3. If Δ < 0, the equation has no real roots.
Substituting the values into the discriminant formula, we have:
Δ = (k-3)^2 - 4(2)(k-5)
Now, let's solve for Δ and determine the value of k:
Δ = (k-3)^2 - 8(k-5)
Δ = k^2 - 6k + 9 - 8k + 40
Δ = k^2 - 14k + 49 - 8k
Simplifying further, we have:
Δ = k^2 - 22k + 49
Now, we can evaluate the discriminant to determine the number of real roots:
1. If Δ > 0:
For two distinct real roots, Δ > 0. Therefore, we need to solve the inequality:
k^2 - 22k + 49 > 0
2. If Δ = 0:
For one real root, Δ = 0. Therefore, we need to solve the equation:
k^2 - 22k + 49 = 0
3. If Δ < 0:
For no real roots, Δ < 0. Therefore, we need to solve the inequality:
k^2 - 22k + 49 < 0
By solving these equations and inequalities, we can find the values of k that correspond to each case.
To determine the number of real roots for the equation 2X^2 + (k-3)x + k-5 = 0, we need to analyze the discriminant. The discriminant of a quadratic equation in the form ax^2 + bx + c = 0 is given by b^2 - 4ac.
1. If the discriminant is positive (b^2 - 4ac > 0), the equation has two distinct real roots.
2. If the discriminant is zero (b^2 - 4ac = 0), the equation has one real root (repeated root).
3. If the discriminant is negative (b^2 - 4ac < 0), the equation has no real roots.
Let's apply this knowledge to the given equation 2X^2 + (k-3)x + k-5 = 0.
The coefficient of x^2 is 2 (a), the coefficient of x is (k-3) (b), and the constant term is (k-5) (c).
Now, substitute these values into the discriminant formula: (k-3)^2 - 4(2)(k-5).
Simplifying this expression yields (k^2 - 6k + 9) - 8(k-5).
Expanding further gives k^2 - 6k + 9 - 8k + 40.
Combining like terms, the expression reduces to k^2 - 14k + 49.
Now, we can determine the number of real roots based on the discriminant.
1. If the discriminant (k^2 - 14k + 49) is positive, the equation has two distinct real roots.
2. If the discriminant is zero, the equation has one real root (repeated root).
3. If the discriminant is negative, the equation has no real roots.
I did this yesterday
http://www.jiskha.com/display.cgi?id=1452172535
For the 2nd and third part set b^2 - 4ac > 0 and then < 0 for two real roots and and no real root respectively.