Question

2-3: factored form home... part 2 of 2 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 applying the zero product property. b. the student should have evaluated the function 2x² + 7x + 5 with x = 0 to find the zeros of the equation. 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 contradiction 0 ≠ 5 in the fourth line means there are no real solutions. what is the correct solution? x = (use a comma to separate answers as needed.)

Explanation:

Step1: Factor the quadratic equation

We have $0 = 2x^{2}+7x + 5$. We need to find two numbers that multiply to $2\times5=10$ and add up to $7$. The numbers are $2$ and $5$. So, $0=2x^{2}+2x + 5x+5=2x(x + 1)+5(x + 1)=(2x + 5)(x+1)$.

Step2: Apply the Zero - Product Property

The Zero - Product Property states that if $ab = 0$, then either $a = 0$ or $b = 0$. So, if $(2x + 5)(x + 1)=0$, then $2x+5=0$ or $x + 1=0$.
For $2x+5=0$, we solve for $x$:
$2x=-5$, so $x=-\frac{5}{2}$.
For $x + 1=0$, we solve for $x$:
$x=-1$.
The error in the student's work is that when applying the Zero - Product Property to $0=(2x + 5)(x + 1)$, the student made an incorrect split. The correct way is to set each factor equal to zero, not set parts of the factored form in an incorrect way as the student did.

Answer:

$x=-\frac{5}{2},-1$