For which equasion does the sum of the roots equal 3/4 and the product of the roots equal -2?
(a)4x^2-8x+3=0
(b)4x^2+8x+3=0
(c)4x^2-3x-8=0
(d)4x^2+3x-2=0
For the general quadratic equation
ax²+bx+c=0
the sum of the roots is -b/a,
and the product of the roots is c/a.
So we look for an equation where
-b/a = 3/4 and
c/a = -2
(a) and (b) are rejected because c/a>0
for (c),
c/a = -8/4 = -2,
-b/a = -(-3)/4 = 3/4
So the answer is (c)
Well, I don't mean to "equa-tion" or anything, but let's solve this with a bit of humor!
The sum of the roots for an equation of the form ax^2 + bx + c = 0 is given by -b/a, while the product of the roots is given by c/a.
Now, let's put on our silly math hats and do some calculations!
(a) For 4x^2 - 8x + 3 = 0, the sum of the roots is -(-8)/4 = 2 and the product of the roots is 3/4. So, this equation doesn't match our given conditions.
(b) For 4x^2 + 8x + 3 = 0, the sum of the roots is -8/4 = -2 and the product of the roots is 3/4. Oops, this option doesn't meet the requirements either.
(c) For 4x^2 - 3x - 8 = 0, the sum of the roots is -(-3)/4 = 3/4 and the product of the roots is -8/4 = -2. Hurray, we have a match!
(d) For 4x^2 + 3x - 2 = 0, the sum of the roots is -3/4 and the product of the roots is -2/4 = -1/2. Nope, this equation doesn't make the cut either.
So, the equation that satisfies both conditions is (c) 4x^2 - 3x - 8 = 0. Happy solving!
To find the equation for which the sum of the roots is 3/4 and the product of the roots is -2, we can use the fact that the sum of the roots of a quadratic equation is given by the formula:
Sum of roots = -b/a
where 'a' is the coefficient of x^2 and 'b' is the coefficient of x.
Similarly, the product of the roots is given by the formula:
Product of roots = c/a
where 'c' is the constant term.
Let's calculate the sum and product of the roots for each equation and see which equation satisfies the given conditions:
(a) 4x^2 - 8x + 3 = 0:
Sum of roots = -(-8)/4 = 8/4 = 2
Product of roots = 3/4
(b) 4x^2 + 8x + 3 = 0:
Sum of roots = -(8)/4 = -2
Product of roots = 3/4
(c) 4x^2 - 3x - 8 = 0:
Sum of roots = -(-3)/4 = 3/4
Product of roots = -8/4 = -2
(d) 4x^2 + 3x - 2 = 0:
Sum of roots = -(3)/4 = -3/4
Product of roots = -2/4 = -1/2
Based on the calculations, we can conclude that the equation (c) 4x^2 - 3x - 8 = 0 is the correct equation for which the sum of the roots is 3/4 and the product of the roots is -2.
To find an equation that satisfies the given conditions, we can use the fact that for a quadratic equation of the form ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.
Let's analyze the given conditions:
- The sum of the roots is 3/4.
- The product of the roots is -2.
For option (a) 4x^2-8x+3=0:
- The sum of the roots would be -(-8)/4 = 2.
- The product of the roots would be 3/4.
Since the sum and the product of the roots do not match the given conditions, we can eliminate option (a).
For option (b) 4x^2+8x+3=0:
- The sum of the roots would be -8/4 = -2.
- The product of the roots would be 3/4.
Since neither the sum nor the product of the roots matches the given conditions, we can eliminate option (b).
For option (c) 4x^2-3x-8=0:
- The sum of the roots would be -(-3)/4 = 3/4, which matches the given condition for the sum of the roots.
- The product of the roots would be -8/4 = -2, which matches the given condition for the product of the roots.
The equation 4x^2-3x-8=0 satisfies both the given conditions, so the correct choice is option (c).
Therefore, for the equation to have a sum of roots equal to 3/4 and a product of roots equal to -2, the equation is 4x^2-3x-8=0.