Given the roots, 1/2 and 4, find:
(A) The quadratic equation
2x^2-9x+4 ?
(B) The sum of the roots
4 1/2, or 9/2
(C) The product of the roots
2
Are these answers correct?
Yes, all correct!
Thanks again!
(A) The quadratic equation can be found by factoring the given roots:
The roots are 1/2 and 4.
To find the equation, we can write it in the form: (x - root1)(x - root2) = 0
Plugging in the given roots, we have:
(x - 1/2)(x - 4) = 0
Expanding this equation gives: x^2 - (4 + 1/2)x + (1/2)(4) = 0
Simplifying further, we get: x^2 - (9/2)x + 2 = 0
So, the quadratic equation is 2x^2 - 9x + 4.
(B) The sum of the roots can be found by adding the roots together:
Sum of the roots = root1 + root2
Sum of the roots = 1/2 + 4 = 9/2
So, the sum of the roots is 9/2.
(C) The product of the roots can be found by multiplying the roots together:
Product of the roots = root1 * root2
Product of the roots = (1/2) * 4 = 2
So, the product of the roots is 2.
Yes, these answers are correct.
Yes, these answers are correct.
To find the quadratic equation given the roots, you can start by using the fact that the roots of a quadratic equation are the values of x that make the equation equal to zero. In this case, the roots are 1/2 and 4.
Using the roots, you can set up the equations:
x = 1/2 (because 1/2 is a root)
x = 4 (because 4 is a root)
To find the quadratic equation, you can start by setting up two binomial factors:
(x - root1)(x - root2) = 0
(x - 1/2)(x - 4) = 0
Next, you can expand this equation:
x^2 - 4x - (1/2)x + 2 = 0
x^2 - (4 + 1/2)x + 2 = 0
x^2 - (9/2)x + 2 = 0
So, the quadratic equation with the given roots is 2x^2 - 9x + 4.
To find the sum of the roots, you simply add the two roots:
1/2 + 4 = 9/2
The sum of the roots is 9/2 or 4 1/2.
To find the product of the roots, you multiply the two roots:
1/2 * 4 = 2
The product of the roots is 2.