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 \(b^2 - 4ac\):
-
Calculate \(b^2\): \[ (-0.7)^2 = 0.49 \]
-
Calculate \(4ac\): \[ 4(0.1)(1.2) = 0.48 \]
-
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:
- \(x = \frac{0.8}{0.2} = 4\)
- \(x = \frac{0.6}{0.2} = 3\)
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