Factor $x^2-2 x+3$
A. $(x-3)(x-1)$
B. $(x+3)(x+1)$
C. $(x-3)(x+1)$
D. Prime
Answer (1)
The problem requires us to factor the quadratic expression x 2 − 2 x + 3 . We calculate the discriminant Δ = b 2 − 4 a c = ( − 2 ) 2 − 4 ( 1 ) ( 3 ) = − 8 . Since the discriminant is negative, the quadratic expression has no real roots and is therefore prime. The final answer is P r im e .
Explanation
Problem Analysis We are asked to factor the quadratic expression x 2 − 2 x + 3 . We will determine if the quadratic expression can be factored by computing the discriminant.
Calculating the Discriminant The discriminant, denoted as Δ , is calculated using the formula Δ = b 2 − 4 a c , where a , b , and c are the coefficients of the quadratic expression a x 2 + b x + c . In our case, a = 1 , b = − 2 , and c = 3 .
Evaluating the Discriminant Substituting the values of a , b , and c into the discriminant formula, we get: Δ = ( − 2 ) 2 − 4 ( 1 ) ( 3 ) = 4 − 12 = − 8 Since the discriminant is negative, the quadratic expression has no real roots.
Conclusion If the discriminant is negative, the quadratic expression cannot be factored using real numbers. Therefore, the quadratic expression x 2 − 2 x + 3 is prime.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to design efficient algorithms. Understanding how to factor quadratic expressions allows us to solve problems related to optimization, modeling, and prediction in various fields.
