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
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
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
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.