Solve each equation by factoring
v(squared) - 10v + 21 = 0
v^2-10v+21=v^2-7v-3v+21
=v(v-7)-3(v-7)
=(v-7)(v-3)
To solve the equation v^2 - 10v + 21 = 0 by factoring, we need to rewrite the equation in the form of (v - x)(v - y) = 0, where x and y are the factors.
Step 1:
Set up two sets of parentheses: (v - x)(v - y) = 0.
Step 2:
Find two numbers, x and y, such that their product is equal to the constant term (21) and their sum is equal to the coefficient of the middle term (-10).
In this case, we need to find numbers x and y such that xy = 21 and x + y = -10.
Step 3:
To find the values of x and y, we can set up a system of equations:
xy = 21
x + y = -10
Step 4:
Now, we need to solve the system of equations. By inspection, we can see that -3 and -7 are such numbers:
(-3) * (-7) = 21 (product is 21)
(-3) + (-7) = -10 (sum is -10)
So, we can rewrite the equation as:
(v - 3)(v - 7) = 0
Step 5:
Now we set each factor equal to zero and solve for v:
v - 3 = 0 → v = 3
v - 7 = 0 → v = 7
Hence, the solutions to the equation v^2 - 10v + 21 = 0 are v = 3 and v = 7.