Determine the real roots of the equation: (5x^2 + 20)(3x^2 - 48) = 0
the first factor 5x^2 + 20) = 0 will not yield any real roots, so
3x^2 - 48 = 0
x^2 = 16
x = ± 4
To determine the real roots of the equation (5x^2 + 20)(3x^2 - 48) = 0, we need to set each factor equal to zero and solve for x.
Setting 5x^2 + 20 = 0:
5x^2 + 20 = 0
5x^2 = -20
x^2 = -20/5
x^2 = -4
x = ±√(-4)
Since the square root of a negative number is not a real number, there are no real roots for this factor.
Setting 3x^2 - 48 = 0:
3x^2 - 48 = 0
3x^2 = 48
x^2 = 48/3
x^2 = 16
x = ±√16
x = ±4
The real roots of the equation (5x^2 + 20)(3x^2 - 48) = 0 are x = 4 and x = -4.
To determine the real roots of the equation (5x^2 + 20)(3x^2 - 48) = 0, we need to solve each factor independently and find the values of x that make each factor equal to zero. By doing this, we can identify the real roots of the equation.
First, let's solve the factor 5x^2 + 20 = 0 for x:
5x^2 + 20 = 0
Divide both sides of the equation by 5:
x^2 + 4 = 0
Subtract 4 from both sides:
x^2 = -4
Take the square root of both sides (remembering there is a positive and negative square root):
x = ±√(-4)
Here, we encounter a problem since we are looking for real roots, meaning values of x where the square root is a real number. However, the square root of a negative number (√(-4)) is not a real number. Therefore, there are no real roots for the factor 5x^2 + 20 = 0.
Next, let's solve the factor 3x^2 - 48 = 0 for x:
3x^2 - 48 = 0
Add 48 to both sides of the equation:
3x^2 = 48
Divide both sides of the equation by 3:
x^2 = 16
Take the square root of both sides:
x = ±√16
x = ±4
In this case, we have two real roots: x = 4 and x = -4.
Therefore, the real roots of the equation (5x^2 + 20)(3x^2 - 48) = 0 are x = 4 and x = -4.