Question

solve each of the following equations using the principle of zero products and match it to its correct pair of solutions on the right.each pair on the right will be used once.
(x-5)(2x+1)=0
(x+5)(2x-1)=0
(5x+1)(x-2)=0
(5x-1)(x+2)=0
x=2 or x=-1/5
x=-2 or x=1/5
x=-5 or x=1/2
x=5 or x=-1/2

Answer 1

To solve each equation using the principle of zero products, we set each factor equal to zero and solve for \(x\).

1. **Equation**: \((x - 5)(2x + 1) = 0\)

- Set each factor to zero:
- \(x - 5 = 0 \Rightarrow x = 5\)
- \(2x + 1 = 0 \Rightarrow 2x = -1 \Rightarrow x = -\frac{1}{2}\)

- **Solutions**: \(x = 5\) or \(x = -\frac{1}{2}\)

2. **Equation**: \((x + 5)(2x - 1) = 0\)

- Set each factor to zero:
- \(x + 5 = 0 \Rightarrow x = -5\)
- \(2x - 1 = 0 \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}\)

- **Solutions**: \(x = -5\) or \(x = \frac{1}{2}\)

3. **Equation**: \((5x + 1)(x - 2) = 0\)

- Set each factor to zero:
- \(5x + 1 = 0 \Rightarrow 5x = -1 \Rightarrow x = -\frac{1}{5}\)
- \(x - 2 = 0 \Rightarrow x = 2\)

- **Solutions**: \(x = 2\) or \(x = -\frac{1}{5}\)

4. **Equation**: \((5x - 1)(x + 2) = 0\)

- Set each factor to zero:
- \(5x - 1 = 0 \Rightarrow 5x = 1 \Rightarrow x = \frac{1}{5}\)
- \(x + 2 = 0 \Rightarrow x = -2\)

- **Solutions**: \(x = -2\) or \(x = \frac{1}{5}\)

Now let's match each set of solutions to the pairs given on the right:

- **(x - 5)(2x + 1) = 0** gives **\(x = 5\) or \(x = -\frac{1}{2}\)**.
- **(x + 5)(2x - 1) = 0** gives **\(x = -5\) or \(x = \frac{1}{2}\)**.
- **(5x + 1)(x - 2) = 0** gives **\(x = 2\) or \(x = -\frac{1}{5}\)**.
- **(5x - 1)(x + 2) = 0** gives **\(x = -2\) or \(x = \frac{1}{5}\)**.

Matching results:
1. **(x - 5)(2x + 1) = 0** → **\(x = 5\) or \(x = -\frac{1}{2}\)**
2. **(x + 5)(2x - 1) = 0** → **\(x = -5\) or \(x = \frac{1}{2}\)**
3. **(5x + 1)(x - 2) = 0** → **\(x = 2\) or \(x = -\frac{1}{5}\)**
4. **(5x - 1)(x + 2) = 0** → **\(x = -2\) or \(x = \frac{1}{5}\)**

In summary:
- (x - 5)(2x + 1) = 0 → **x = 5 or x = -1/2**
- (x + 5)(2x - 1) = 0 → **x = -5 or x = 1/2**
- (5x + 1)(x - 2) = 0 → **x = 2 or x = -1/5**
- (5x - 1)(x + 2) = 0 → **x = -2 or x = 1/5**