When factoring an equation, how do I know if the result is prime? For example, I have a problem x^2 - 2x + 4 and I think it is prime because I can't solve it. Am I right?
If an expression cannot be factored over the rational numbers, it is often considered "prime"
since x^2 - 2x + 4 cannot be factored with rational factors, you are correct.
To determine if a factored equation is prime, you need to follow these steps:
1. Factorize the equation: For your example, x^2 - 2x + 4, you can see that it cannot be factored further using integers since the quadratic equation does not have any real roots.
2. Verify if the equation is prime: If you cannot factorize the equation any further, you can check if it is prime by making use of a mathematical property. In this case, you can use the discriminant of the quadratic equation.
The discriminant, represented as "D," is calculated using the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In the given equation x^2 - 2x + 4, the coefficients are a = 1, b = -2, and c = 4. Therefore, the discriminant is:
D = (-2)^2 - 4(1)(4)
= 4 - 16
= -12
If the discriminant is negative, like in this case, it means that the equation is prime since it does not have any real roots.
Hence, in your example, x^2 - 2x + 4 is prime because it cannot be further factored and its discriminant is negative.
To determine if the factored equation is prime, you should check if it can be factored further. In this case, let's factor the equation x^2 - 2x + 4 and see if it can be simplified.
To factor the quadratic equation, you can use different methods, such as factoring by grouping, completing the square, or using the quadratic formula. In this example, we will use factoring by grouping.
1. Start with the equation: x^2 - 2x + 4.
2. Look for two numbers whose product is equal to the constant term (4) and whose sum is equal to the coefficient of the linear term (-2). In this case, those numbers are -2 and -2 since -2 * -2 = 4 and -2 + -2 = -4.
3. Rewrite the equation using these numbers: x^2 - 2x - 2x + 4.
4. Group the terms: (x^2 - 2x) + (-2x + 4).
5. Factor out the greatest common factor from each grouped pair: x(x - 2) - 2(x - 2).
6. Notice that (x - 2) is a common factor in both terms.
7. Combine the factored terms: (x - 2)(x - 2) or (x - 2)^2.
The factored form is (x - 2)^2. In this case, the equation is not prime because it can be factored further into a perfect square.