- The quadratic expression \(3x^2 + 17x - 6\) is factored by grouping as \((3x - 1)(x + 6)\).
- Setting each factor to zero, we get \(3x - 1 = 0\) and \(x + 6 = 0\).
- Solving these simple equations yields \(x = \frac{1}{3}\) and \(x = -6\).
- The correct solutions are \(\boxed{x = \frac{1}{3}, x = -6}\).
Explanation
- Analyzing the Equation and Factorization
The problem requires us to solve the quadratic equation \(3x^2 + 17x - 6 = 0\) by verifying the factorization provided by a student. A quadratic equation in the form \(ax^2 + bx + c = 0\) can often be solved by factoring the quadratic expression into the form \((px + q)(rx + s) = 0\), and then applying the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
- Verifying the Factoring Steps
The student correctly broke down the middle term \(17x\) into \(18x - x\) to facilitate grouping:
\[3x^2 + 17x - 6 = 3x^2 + 18x - x - 6\]
Grouping the terms, we have:
\[(3x^2 + 18x) - (x + 6) = 3x(x + 6) - 1(x + 6)\]
Factoring out the common binomial \((x + 6)\), we obtain:
\[(3x - 1)(x + 6) = 0\]
This confirms the student's factored expression is correct.
- Identifying the Correct Solutions
To find the solutions, we set each factor equal to zero:
- \(3x - 1 = 0 \Rightarrow 3x = 1 \Rightarrow x = \frac{1}{3}\)
- \(x + 6 = 0 \Rightarrow x = -6\)
Thus, the possible solutions are \(x = \frac{1}{3}\) and \(x = -6\). Comparing these with the provided options, we see that \(x = \frac{1}{3}\) and \(x = -6\) are indeed the correct solutions.
Examples
Quadratic equations are fundamental in modeling physical systems where acceleration is constant, such as calculating the time an object stays in the air when thrown upwards. For instance, if the height of a projectile is given by \(h(t) = -5t^2 + 20t + 1\), setting \(h(t) = 0\) helps determine when the object hits the ground. Learning how to factor and solve these equations is a crucial skill for analyzing motion, predicting outcomes in economics, and designing efficient structures in engineering.