Question

Pedro and Rebecca were trying to solve the equation:
\[0.1x^2-0.7x+1.2=0\]
Pedro said, “I can multiply the entire equation by
\[10\] and then factor. Then I can solve using the zero product property.”
Rebecca said, “I'll solve using the quadratic formula with
\[a=0.1\],
\[b=-0.7\], and
\[c=1.2\].”
Whose solution strategy would work?
Choose 1 answer:
Choose 1 answer:
(Choice A) Only Pedro's
A
Only Pedro's
(Choice B) Only Rebecca's
B
Only Rebecca's
(Choice C) Both
C
Both
(Choice D) Neither
D
Neither

Answer 1

To determine whose solution strategy would work, let's analyze both approaches.

**Pedro's Approach:**
Pedro proposes to multiply the entire equation by \(10\), resulting in:

\[
10(0.1x^2) - 10(0.7x) + 10(1.2) = 0
\]
This simplifies to:

\[
x^2 - 7x + 12 = 0
\]

Now, Pedro can factor the left-hand side of the equation:

\[
(x - 3)(x - 4) = 0
\]

This leads to the solutions:

\[
x - 3 = 0 \Rightarrow x = 3
\]
\[
x - 4 = 0 \Rightarrow x = 4
\]

This indicates that Pedro's approach works.

**Rebecca's Approach:**
Rebecca intends to use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Given \(a = 0.1\), \(b = -0.7\), and \(c = 1.2\), we can compute \(b^2 - 4ac\):

1. Calculate \(b^2\):
\[
(-0.7)^2 = 0.49
\]

2. Calculate \(4ac\):
\[
4(0.1)(1.2) = 0.48
\]

3. Now, calculate \(b^2 - 4ac\):
\[
0.49 - 0.48 = 0.01
\]

Now, let's find \(x\) using the quadratic formula:

\[
x = \frac{-(-0.7) \pm \sqrt{0.01}}{2(0.1)}
= \frac{0.7 \pm 0.1}{0.2}
\]

This yields:

1. \(x = \frac{0.8}{0.2} = 4\)
2. \(x = \frac{0.6}{0.2} = 3\)

Rebecca's approach also works and leads to the same solutions, \(x = 3\) and \(x = 4\).

**Conclusion:**
Both Pedro and Rebecca's methods provide valid solutions to the original equation. Therefore, the correct choice is:

**(Choice C) Both**