Question

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
x - 1 = x
(use a comma to separate answers as needed)
what is the correct solution?
x =

options:
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 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:

Step1: Factor the quadratic expression

Given the equation \(0 = 2x^{2}+7x + 5\), we can factor it by grouping. First, rewrite \(7x\) as \(2x+5x\), so the equation becomes \(0=2x^{2}+2x + 5x+5\). Then factor by grouping: \(0 = 2x(x + 1)+5(x + 1)\), and further factor to get \(0=(2x + 5)(x + 1)\).

Step2: Apply the Zero Product Property

Using the Zero Product Property, if \((2x + 5)(x + 1)=0\), then either \(2x+5 = 0\) or \(x + 1=0\).

Step3: Solve for \(x\) from each factor

• For \(2x+5 = 0\): Subtract 5 from both sides: \(2x=-5\), then divide by 2: \(x=-\frac{5}{2}\).
• For \(x + 1=0\): Subtract 1 from both sides: \(x=-1\).

Answer:

\(x = -1, -\frac{5}{2}\) (or \(x=-1, -2.5\))