Use factoring and the zero-product property to solve the following problems.
1. z(z-1)(z+3)=0
2. x^2-x-10=2
3. 4a^2-11a+6=0
Sure! Let's solve each problem using factoring and the zero-product property:
1. z(z-1)(z+3)=0
To solve this equation, we'll 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 to zero and solve for z:
z = 0 or z-1 = 0 or z+3 = 0
For the first factor, z = 0.
For the second factor, z-1 = 0, so adding 1 to both sides, we get z = 1.
For the third factor, z+3 = 0, so subtracting 3 from both sides, we get z = -3.
Therefore, the solutions to the equation are z = 0, z = 1, and z = -3.
2. x^2-x-10=2
To solve this equation, we'll move all the terms to one side to set it equal to zero:
x^2 - x - 10 - 2 = 0
x^2 - x - 12 = 0
Now, we'll factorize the quadratic expression:
(x - 4)(x + 3) = 0
To apply the zero-product property, we set each factor to zero and solve for x:
x - 4 = 0, so x = 4
x + 3 = 0, so x = -3
Therefore, the solutions to the equation are x = 4 and x = -3.
3. 4a^2-11a+6=0
To solve this equation, we'll start by factoring the quadratic expression:
(4a - 3)(a - 2) = 0
Now, we'll set each factor to zero and solve for a:
4a - 3 = 0, so adding 3 to both sides, we get 4a = 3, and dividing both sides by 4, we obtain a = 3/4.
a - 2 = 0, so adding 2 to both sides, we get a = 2.
Therefore, the solutions to the equation are a = 3/4 and a = 2.
Using factoring and the zero-product property allows us to find the solutions to these equations by breaking them down into simpler factors and setting them equal to zero.