Question

describe and correct the error a student made in solving a quadratic equation.
0 = 2x² + 7x + 5
0 = 2x² + 2x + 5x + 5
0 = 2x(x + 1) + 5(x + 1)
0 = 2x, 0 = x + 1, 0 ≠ 5
0 = x, -1 = x
choose the best explanation of the error below.
a. the contradiction 0 ≠ 5 in the fourth line means there are no real solutions.
b. the student should have completed the factoring and rewritten 2x(x + 1) + 5(x + 1) as (2x + 5)(x + 1) before applying the zero product property.
c. in the first step, the student should have found factors of 5 that add to give 7 in order to factor the quadratic expression on the right.
d. the student should have evaluated the function 2x² + 7x + 5 with x = 0 to find the zeros of the equation.

Explanation:

To solve the quadratic equation by factoring, after getting \(0 = 2x(x + 1)+5(x + 1)\), the student should factor out \((x + 1)\) to get \(0=(2x + 5)(x + 1)\) (completing the factoring) before applying the Zero Product Property. Option A is wrong as \(0
eq5\) is just a term, not a contradiction for solutions. Option C is wrong as the coefficient of \(x^2\) is 2, not 1. Option D is wrong as evaluating at \(x = 0\) isn't the way to find zeros here. So the correct explanation is B.

Answer:

B. The student should have completed the factoring and rewritten \(2x(x + 1)+5(x + 1)\) as \((2x + 5)(x + 1)\) before applying the Zero Product Property.