Use factoring and the zero-product property to solve the following problems:
z(z - 1)(z + 3) = 0
x^2 - x - 10 = 2
4a^ - 11a + 6 = 0
9r^r - 30r + 21 = -4
Please show work. Thank you so much!
the first one is already factored, so how come you are stuck on it ?
2nd:
get the equation = 0
then it factors quite easily
4a^2 - 11a + 6 = 0 and 9r^r - 30r + 21 = -4
have you not been taught some kind of a method to factor these kind of trinomials?
A common method these days is "decomposition"
Thanks. :) I think I got it now. You rock.
Sure! I'll show you step-by-step how to solve each of these problems using factoring and the zero-product property.
1. z(z - 1)(z + 3) = 0:
To solve this equation, we can use the zero-product property, which states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. So we set each factor equal to zero:
z = 0,
z - 1 = 0,
z + 3 = 0.
Now solve each equation separately:
z = 0,
z - 1 = 0 --> z = 1,
z + 3 = 0 --> z = -3.
So the solutions to the equation z(z - 1)(z + 3) = 0 are z = 0, z = 1, and z = -3.
2. x^2 - x - 10 = 2:
First, move all the terms to one side to get the equation equal to zero:
x^2 - x - 10 - 2 = 0,
x^2 - x - 12 = 0.
Next, we need to factor the quadratic equation:
(x - 4)(x + 3) = 0.
Now set each factor equal to zero:
x - 4 = 0 --> x = 4,
x + 3 = 0 --> x = -3.
So the solutions to the equation x^2 - x - 10 = 2 are x = 4 and x = -3.
3. 4a^2 - 11a + 6 = 0:
Again, move all the terms to one side:
4a^2 - 11a + 6 - 0 = 0,
4a^2 - 11a + 6 = 0.
Next, we factor the quadratic:
(4a - 3)(a - 2) = 0.
Now set each factor equal to zero:
4a - 3 = 0 --> 4a = 3 --> a = 3/4,
a - 2 = 0 --> a = 2.
So the solutions to the equation 4a^2 - 11a + 6 = 0 are a = 3/4 and a = 2.
4. 9r^2 - 30r + 21 = -4:
Move all the terms to one side:
9r^2 - 30r + 21 + 4 = 0,
9r^2 - 30r + 25 = 0.
Now, let's factor the quadratic:
(3r - 5)(3r - 5) = 0 --> (3r - 5)^2 = 0.
Set the factor equal to zero:
3r - 5 = 0 --> 3r = 5 --> r = 5/3.
So the solution to the equation 9r^2 - 30r + 21 = -4 is r = 5/3.
I hope this explanation helps! Let me know if you have any other questions.