How would you solve for x in this equation:
(x^2-9)(2x-1)=0
For the problem, a value of x which makes a factor 0 is a solution.
(x^2-9)(2x-1) = 0
factoring (x^2-9) further
(x+3)(x-3)(2x-1) = 0
For each factor, what value of x will make the term 0?
The first term, (x+3), will be 0 for x=-3.
The second term is 0 for x=?
The third term is 0 for x=?
To solve for x in the equation (x^2 - 9)(2x - 1) = 0, we can use the zero product property. According to this property, if the product of two or more factors is zero, then at least one of the factors must be zero. Therefore, we need to set each factor equal to zero and solve for x.
Setting the first factor, x^2 - 9, equal to zero gives us:
x^2 - 9 = 0
To solve this quadratic equation, we can apply the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). Rewriting the equation using this formula, we have:
(x + 3)(x - 3) = 0
Now, we set the second factor, 2x - 1, equal to zero:
2x - 1 = 0
To solve for x, we can isolate x by adding 1 to both sides of the equation:
2x = 1
Next, divide both sides by 2 to find the value of x:
x = 1/2
So, we have two possibilities for x: x = 1/2 and x = -3 or x = 3.
Hence, the solution to the equation (x^2 - 9)(2x - 1) = 0 is x = 1/2, x = -3, and x = 3.
To solve the equation (x^2 - 9)(2x - 1) = 0 for x, we can use the zero product property. According to this property, if two factors multiply together to give a product of zero, then at least one of the factors must be equal to zero.
So, we set each factor equal to zero and solve for x:
Factor 1: x^2 - 9 = 0
To solve this quadratic equation, we can factor it by recognizing it as a difference of squares.
x^2 - 9 = (x - 3)(x + 3)
Now, apply the zero product property to set each factor equal to zero and solve for x:
x - 3 = 0 or x + 3 = 0
For x - 3 = 0, add 3 to both sides:
x = 3
For x + 3 = 0, subtract 3 from both sides:
x = -3
So, the solutions for the first factor are x = 3 and x = -3.
Factor 2: 2x - 1 = 0
To solve for x, isolate the variable:
2x = 1
Divide both sides by 2:
x = 1/2
So, the solution for the second factor is x = 1/2.
Therefore, the equation (x^2 - 9)(2x - 1) = 0 has three solutions: x = 3, x = -3, and x = 1/2.