If a-b=3,ab=7,then(a+b)is _______
b=7/a
a-7/a=3
a^2-7=3a
a^2-3a-7=0
a= (3+-sqrt(9+28)/2= ....
a+b=a+7/a= you do it.
or,
(a-b)^2 = a^2 - 2ab + b^2 = a^2 + b^2 - 14 = 9
so, a^2+b^2 = 23
(a+b)^2 = a^2+2ab+b^2 = 23+14 = 37
(a+b) = √37
To find the value of (a+b), we can use the given equations. Let's solve it step by step.
Given:
a - b = 3 ---(Equation 1)
ab = 7 ---(Equation 2)
Step 1: Solve Equation 1 for a in terms of b:
a = b + 3
Step 2: Substitute the value of a from Step 1 into Equation 2:
(b + 3)b = 7
Step 3: Simplify the equation:
b^2 + 3b = 7
Step 4: Rearrange the equation into a quadratic form:
b^2 + 3b - 7 = 0
Step 5: Use the quadratic formula to solve for b:
b = (-3 ± √(3^2 - 4*1*(-7)))/(2*1)
Simplifying further:
b = (-3 ± √(9 + 28))/2
b = (-3 ± √37)/2
Step 6: Substitute the value of b back into Equation 1 to find a:
a = b + 3
Using the positive square root of 37, we get:
a = (-3 + √37)/2
Step 7: Calculate (a+b):
(a + b) = ((-3 + √37)/2) + ((-3 + √37)/2)
= -3/2 + √37/2 - 3/2 + √37/2
= -6/2 + 2√37/2
= (-6 + 2√37)/2
Simplifying further, we get:
(a + b) = -3 + √37
Therefore, (a + b) is -3 + √37 when a - b = 3 and ab = 7.