Consider the quadratic equation
m^2 x^2 + 2(2m-5)x + 8 = 0
a) If 2 is a root of the equation, find the possible values of m (there are four possible values)
b) For each value of m, find the other root
I have actually calculated the answer m=1 or m=-3 for (a), but I was told that there are four possible values of m. I was stuck on this part.
since 2 is a root,
4m^2 + 2(2m-5)(2) + 8 = 0
4m^2 + 8m - 20 + 8 = 0
4m^2 + 8m - 12 =
m^2 + 2m - 3 = 0
(m+3)(m-1) = 0
m = -3, or m = 1
if m = -3, we have
9x^2 -22x + 8 = 0
(x-2)(9x - 4) = 0
x = 2 (our given) or x = 4/9
if m = 1 , we have
x^2 -6x + 8 = 0
(x-2)(x-4) = 0
x=2 or x=4
Like you, I only get 2 values for m
and each value of m yields 2 answers. (one doubling up)
I also approached the problem using the sum and product of roots property.
let the roots be a and b, (we have a quadratic in x)
but we know one of them let b = 2
sum of roots = a+2
product of roots = 2a
sum of roots = -2(2m-5)/m^2
a+2 = (-4m + 10)/m^2 - 2
a = (-4m +10 - 2m^2)/m^2
product of roots = 8/m^2
2a = 8/m^2
a = 4/m^2
4/m^2 = (-4m + 10 - 2m^2)/m^2
4 = -4m + 10 - 2m^2
2m^2 + 4m - 6 = 0
m^2 + 2m - 3 = 0
(m+3)(m-1) = 0
m = -3 or m = 1
same as above, and only 2 values of m
To find the possible values of m when 2 is a root of the given quadratic equation, we can use the fact that if a number is a root of a quadratic equation, then the equation will be equal to zero when that number is substituted.
a) Let's substitute 2 into the given quadratic equation and set it equal to zero:
(m^2)(2^2) + 2(2m - 5)(2) + 8 = 0
Simplifying this equation leads to:
4m^2 + 4(2m - 5) + 8 = 0
Expanding the expression further:
4m^2 + 8m - 20 + 8 = 0
Combining like terms:
4m^2 + 8m - 12 = 0
Now, we have a quadratic equation in terms of m. To find the possible values of m, we can use the quadratic formula, which states that for a quadratic equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Therefore, applying the quadratic formula to our equation:
m = (-8 ± √(8^2 - 4(4)(-12))) / (2*4)
Simplifying further:
m = (-8 ± √(64 + 192)) / 8
m = (-8 ± √(256)) / 8
m = (-8 ± 16) / 8
Simplifying the equations:
m = (8 - 16) / 8 = -8 / 8 = -1
m = (8 + 16) / 8 = 24 / 8 = 3
Therefore, the possible values of m when 2 is a root of the equation are m = -1 and m = 3.
Now let's move on to part b.
b) To find the other root for each value of m, we can use the quadratic formula again. We substitute the calculated values of m into the quadratic equation and solve for x.
When m = -1:
Substituting m = -1 into the given quadratic equation:
(-1)^2x^2 + 2(2(-1) - 5)x + 8 = 0
Simplifying:
x^2 - 4x + 8 = 0
Using the quadratic formula:
x = (-(-4) ± √((-4)^2 - 4(1)(8))) / (2(1))
x = (4 ± √(16 - 32)) / 2
x = (4 ± √(-16)) / 2
Since the square root of a negative number is not defined in the real number system, there are no real solutions for x when m = -1.
When m = 3:
Substituting m = 3 into the given quadratic equation:
(3)^2x^2 + 2(2(3) - 5)x + 8 = 0
Simplifying:
9x^2 + 12x + 8 = 0
Using the quadratic formula:
x = (-12 ± √((12)^2 - 4(9)(8))) / (2(9))
x = (-12 ± √(144 - 288)) / 18
x = (-12 ± √(-144)) / 18
Again, we encounter a square root of a negative number, indicating no real solutions for x when m = 3.
Therefore, for the possible values of m (m = -1 and m = 3), there are no real solutions for the other root of the quadratic equation.