If the numerically smaller root of x2+mx=2 is 3 more than the other one, find the value of m
1 because then the roots will be -2 and 1 1 is numerically smaller then 2 but its greater then -2 by 3.
To find the value of m, we need to solve the given quadratic equation, x^2 + mx = 2.
Let's assume the two roots of the equation are a and b. According to the given information, the numerically smaller root is 3 more than the other one.
So, we can write the following equations:
a = b + 3 (1)
a * b = 2 (2)
To eliminate one variable, let's substitute the value of a from equation (1) into equation (2):
(b + 3) * b = 2
b^2 + 3b - 2 = 0
Now, we have a quadratic equation in terms of b. We can solve it using factoring or the quadratic formula.
By factoring:
(b - 1)(b + 2) = 0
This gives two possible values for b:
b = 1 or b = -2
Now, substitute these values of b back into equation (1) to find the corresponding values for a:
For b = 1:
a = 1 + 3
a = 4
For b = -2:
a = -2 + 3
a = 1
So, we have two possible pairs of roots, (a, b):
(4, 1) and (1, -2).
Now, we can find the value of m. Substitute the values of a and b into the equation x^2 + mx = 2 and see which value of m satisfies both equations.
Using the roots (4, 1):
(4)^2 + m(4) = 2
16 + 4m = 2
4m = -14
m = -14/4
m = -7/2
Using the roots (1, -2):
(1)^2 + m(1) = 2
1 + m = 2
m = 2 - 1
m = 1
Thus, the two possible values of m are -7/2 and 1.
To find the value of m, we need to solve the given equation for x. Let's break down the problem step by step.
Step 1: Formulate the equation
The equation is given as x^2 + mx = 2.
Step 2: Identify the smaller root
Let's assume the smaller root is a.
Step 3: Express the larger root in terms of the smaller root
According to the problem, the numerically smaller root is 3 more than the other root, which means the larger root is a + 3.
Step 4: Set up the equation using the sum and product of roots
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.
In our equation, the sum of the roots is a + (a + 3) = 2a + 3, and the product of the roots is a * (a + 3) = 2.
We can now set up the equation:
Sum of roots = 2a + 3
Product of roots = a(a + 3) = 2
Step 5: Solve the equation
Using the equation from Step 4, we have:
2a + 3 = -m ----- Equation (1)
a(a + 3) = 2 ----- Equation (2)
We can substitute Equation (2) into Equation (1) to solve for m:
2(a + 3) + 3 = -m
2a + 6 + 3 = -m
2a + 9 = -m
To isolate m, multiply both sides by -1:
-m = -2a - 9
Finally, we have the value of m:
m = 2a + 9
Hence, the value of m is 2a + 9.