1.One root of each equation below is the same for both equations. what is this root?
a.3x^4 + 14x^3 + 8x^2=0
b.18x^4 - 50x^2=0
2. Find the other roots of each equation in question 1
a)
x^2(3x^2 + 14x + 8) = 0
x^2 (3x+2)(x+4) = 0
x = 0 , or x=-2/3, or x = -4
b)
2x^2(9x^2 - 25) = 0
2x^2(x+3)(x-3) = 0
x = 0 , x = 3, x = -3
To find the root that is the same for both equations, we need to set the equations equal to each other and solve for x. Let's do that step by step for each equation:
1. Equation a: 3x^4 + 14x^3 + 8x^2 = 0
2. Equation b: 18x^4 - 50x^2 = 0
Step 1: Set the two equations equal to each other:
3x^4 + 14x^3 + 8x^2 = 18x^4 - 50x^2
Step 2: Simplify the equation:
15x^4 - 14x^3 - 58x^2 = 0
Now, to find the common root, we can solve this simplified equation.
To find the other roots of each equation, we will solve them individually.
Let's start with the first equation, a:
1. Equation a: 3x^4 + 14x^3 + 8x^2 = 0
To solve this equation, we can factor out a common term:
x^2(3x^2 + 14x + 8) = 0
Now we have two factors:
x^2 = 0 (Root 1)
3x^2 + 14x + 8 = 0
For the second equation, b:
2. Equation b: 18x^4 - 50x^2 = 0
We can also factor out a common term:
2x^2(9x^2 - 25) = 0
Now we have two factors:
x^2 = 0 (Root 1)
9x^2 - 25 = 0
To find the other roots, we need to solve each of the remaining quadratic equations:
For equation a:
3x^2 + 14x + 8 = 0
We can solve this equation by factoring, completing the square, or using the quadratic formula. The factorization method is the quickest:
(3x + 2)(x + 4) = 0
This gives us two more roots:
3x + 2 = 0 (Root 2)
x + 4 = 0 (Root 3)
Solving these equations gives us:
Root 2: x = -2/3
Root 3: x = -4
For equation b:
9x^2 - 25 = 0
This is a difference of squares, so we can factor it as follows:
(3x - 5)(3x + 5) = 0
This gives us two more roots:
3x - 5 = 0 (Root 2)
3x + 5 = 0 (Root 3)
Solving these equations gives us:
Root 2: x = 5/3
Root 3: x = -5/3
In conclusion:
The common root for both equations is x = 0.
For equation a, the other roots are x = -2/3 and x = -4.
For equation b, the other roots are x = 5/3 and x = -5/3.