Please help me about solving kuadratic equations: X^2-2(1-3m)x+7(3+2m)=0 Find m parametter that equations to have the same solves x1=x2,so Discriminant D=b^2-4ac D=0.
Please Help me.
To find the value of the parameter 'm' for which the equation has equal roots, we will first calculate the discriminant (D) of the quadratic equation. The discriminant is given by the formula D = b^2 - 4ac, where 'a', 'b', and 'c' are the coefficients of the equation.
In this case, the given equation is X^2 - 2(1 - 3m)x + 7(3 + 2m) = 0. Comparing it with the standard form ax^2 + bx + c = 0, we have:
a = 1
b = -2(1 - 3m) = -2 + 6m
c = 7(3 + 2m) = 21 + 14m
Now, substitute the values of 'a', 'b', and 'c' into the discriminant formula:
D = (-2 + 6m)^2 - 4(1)(21 + 14m)
Simplify and expand the equation:
D = 4 - 24m + 36m^2 - 84 - 56m
Combine like terms:
D = 36m^2 - 80m - 80
Now, set the discriminant equal to zero (D = 0) to find the value of 'm' for which the equation has equal roots:
36m^2 - 80m - 80 = 0
To solve this quadratic equation, you can use the quadratic formula or factorization methods. Let's use the quadratic formula:
m = (-b ± √(b^2 - 4ac)) / (2a)
Substitute the values into the quadratic formula:
m = (-(-80) ± √((-80)^2 - 4(36)(-80))) / (2(36))
m = (80 ± √(6400 + 11520)) / 72
m = (80 ± √17920) / 72
Simplify further:
m = (80 ± 10√448) / 72
Finally, simplify and separate the two values of 'm':
m1 = (80 + 10√448) / 72
m2 = (80 - 10√448) / 72
These are the two solutions for 'm' that will make the given quadratic equation have equal roots.