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 student should have evaluated the function 2x² + 7x + 5 with x = 0 to find the zeros of the equation.
b. the contradiction 0 ≠ 5 in the fourth line means there are no real solutions.
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 completed the factoring and rewritten 2x(x + 1) + 5(x + 1) as (2x + 5)(x + 1) before applying the zero product property.

Explanation:

To solve the quadratic equation by factoring, after factoring out the common terms from the grouped parts, we need to factor the entire expression as a product of two binomials. The student factored the quadratic into \(2x(x + 1)+5(x + 1)\) but then incorrectly applied the Zero Product Property to the individual terms \(2x\), \(x + 1\), and \(5\) instead of first combining \(2x\) and \(5\) into a single binomial factor \((2x + 5)\) and then applying the Zero Product Property to \((2x + 5)(x + 1)=0\). Option A is incorrect because evaluating at \(x = 0\) is not the way to find zeros. Option B is incorrect because \(0
eq5\) is not a contradiction that implies no real solutions (the quadratic can still be factored correctly). Option C is incorrect because the student did find factors that add to 7 (2 and 5) when splitting the middle term. Option D correctly identifies the error: the student should have completed the factoring to \((2x + 5)(x + 1)\) before applying the Zero Product Property.

Answer:

D. 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