A student provided the following solution for solving an equation:
In the realm of mathematics, finding the correct solution to an equation is a fundamental skill that students must master. Equations are the backbone of mathematical problem-solving, and they can range from simple linear equations to complex polynomial equations. One such equation that a student recently tackled was a quadratic equation, and they provided a solution that showcased their understanding of the subject matter. This article aims to analyze the student’s solution, discuss its strengths and weaknesses, and provide insights into how it can be improved.
The equation in question was:
x^2 – 5x + 6 = 0
The student’s solution began by factoring the quadratic equation. They correctly identified that the factors of 6 (the constant term) that add up to -5 (the coefficient of the x term) are -2 and -3. Therefore, the student factored the equation as follows:
(x – 2)(x – 3) = 0
Next, the student applied the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. By setting each factor equal to zero, the student derived the two possible solutions for x:
x – 2 = 0
x = 2
x – 3 = 0
x = 3
The student’s solution was correct, and they successfully found the two roots of the quadratic equation: x = 2 and x = 3. However, there are a few areas where the solution could be improved.
Firstly, the student could have included a brief explanation of the zero product property and how it applies to solving quadratic equations. This would help readers understand the reasoning behind the step where the factors are set equal to zero.
Secondly, the student could have expanded on the factoring process. While they correctly identified the factors, they could have mentioned that there are other methods to factor quadratic equations, such as completing the square or using the quadratic formula. This would provide a more comprehensive understanding of the topic.
Lastly, the student could have included a check of their solutions. By substituting the values of x back into the original equation, the student could verify that the solutions are indeed correct.
In conclusion, the student provided a correct solution for solving the quadratic equation x^2 – 5x + 6 = 0. Their use of factoring and the zero product property demonstrated a solid understanding of the subject matter. However, by including additional explanations and expanding on the factoring process, the student could have enhanced the clarity and depth of their solution. This analysis serves as a valuable learning opportunity for students to refine their problem-solving skills and gain a deeper understanding of quadratic equations.
