If 3 is one root of 2x^2 + bx + 3 =0 find the other root and the value of b.
The answers are r2=1/2 and b=-7
but i don't know how to get those.
To determine the other root and the value of b, we can use the fact that the sum of the roots (r1 and r2) of a quadratic equation of the form ax^2 + bx + c = 0 is equal to -b/a, and the product of the roots is equal to c/a.
In this case, the equation is 2x^2 + bx + 3 = 0, and one of the roots is given as 3. Let's call the other root r2 and substitute these values into the formulas:
Sum of the roots:
r1 + r2 = -b/a
Since we know r1 = 3, we can substitute to get:
3 + r2 = -b/2
Product of the roots:
r1 * r2 = c/a
Substituting the values:
3 * r2 = 3/2
Now, we have a system of equations:
3 + r2 = -b/2
3 * r2 = 3/2
To solve this system, we can solve one equation for a variable and substitute it into the other equation.
From the second equation, we can isolate r2:
r2 = (3/2) / 3
r2 = 1/2
Substituting this value back into the first equation:
3 + (1/2) = -b/2
6/2 + 1/2 = -b/2
7/2 = -b/2
To get rid of the denominator, multiply both sides by 2:
7 = -b
Therefore, the other root is r2 = 1/2, and the value of b is -7.