Question

Gisele solved the quadratic equation x^2+5x-14=0
by factoring. Her work is shown below.

Step 1
: x^2+5x-14=0

Step 2
: (x+7)(x-2)=0

Step 3
: x+7=0 , x+2=0
,

Step 4
: x=7
or x=2

Step 5
: {7,2}


Gisele plugged her solutions into the original equation, and found that they did not work.
In which step did she first make a mistake?

Answer 1

Gisele made a mistake in Step 3.

Let's go through her steps again:

1. **Step 1**: She correctly wrote the original equation \(x^2 + 5x - 14 = 0\).

2. **Step 2**: She correctly factored the equation to \((x + 7)(x - 2) = 0\).

3. **Step 3**: Here, Gisele incorrectly wrote the equations from the factored form. Instead of \(x + 7 = 0\) and \(x - 2 = 0\), she mistakenly wrote \(x + 7 = 0\) and \(x + 2 = 0\). The correct equations should have been:
- \(x + 7 = 0\) which gives \(x = -7\)
- \(x - 2 = 0\) which gives \(x = 2\)

Because of this incorrect step, she miscalculated the solutions.

4. **Step 4**: This step arises from her mistaken equations in Step 3 and continues the error.

5. **Step 5**: Resulting in the incorrect solution set \(\{7, 2\}\).

So the first mistake occurred in Step 3 when she incorrectly identified the factors.