To analyze and work with the quadratic expression \(3x^2 + 8x + 4\), we can perform several tasks commonly assigned in a 9th-grade algebra course, such as factoring, finding the vertex, completing the square, or determining the roots using the quadratic formula. Here’s a brief overview of these steps:
1. Factoring:
To factor the quadratic expression \(3x^2 + 8x + 4\), we look for two numbers that multiply to \(3 \times 4 = 12\) (the product of the coefficient of \(x^2\) and the constant term) and add to \(8\) (the coefficient of \(x\)).
The factors of 12 that work are \(6\) and \(2\), so we can rewrite \(8x\) as \(6x + 2x\):
\[ 3x^2 + 6x + 2x + 4 \]
Next, we group the terms:
\[ (3x^2 + 6x) + (2x + 4) \]
Factoring out the common factors from each group gives us:
\[ 3x(x + 2) + 2(x + 2) \]
We can now factor by grouping:
\[ (3x + 2)(x + 2) \]
2. Finding the roots:
We can find the roots (values of \(x\) that make the expression equal to zero) by setting the factored form to zero:
\[ (3x + 2)(x + 2) = 0 \]
This gives us two equations to solve:
- \(3x + 2 = 0\) → \(3x = -2\) → \(x = -\frac{2}{3}\)
- \(x + 2 = 0\) → \(x = -2\)
3. Using the quadratic formula:
The quadratic formula can also be used to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For \(3x^2 + 8x + 4\):
- \(a = 3\)
- \(b = 8\)
- \(c = 4\)
Calculating the discriminant:
\[ b^2 - 4ac = 8^2 - 4(3)(4) = 64 - 48 = 16 \]
Now plug into the quadratic formula:
\[ x = \frac{-8 \pm \sqrt{16}}{2 \cdot 3} = \frac{-8 \pm 4}{6} \]
This gives us two solutions:
- \(x = \frac{-8 + 4}{6} = \frac{-4}{6} = -\frac{2}{3}\)
- \(x = \frac{-8 - 4}{6} = \frac{-12}{6} = -2\)
4. Finding the vertex:
The vertex of a parabola given by the equation \(ax^2 + bx + c\) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting in our values: \[ x = -\frac{8}{2 \cdot 3} = -\frac{8}{6} = -\frac{4}{3} \]
To find the \(y\)-coordinate of the vertex, substitute \(x = -\frac{4}{3}\) back into the original equation:
\[ y = 3\left(-\frac{4}{3}\right)^2 + 8\left(-\frac{4}{3}\right) + 4 = 3\left(\frac{16}{9}\right) - \frac{32}{3} + 4 \]
Calculating: \[ y = \frac{48}{9} - \frac{96}{9} + \frac{36}{9} = \frac{48 - 96 + 36}{9} = \frac{-12}{9} = -\frac{4}{3} \]
So, the vertex is at \(\left(-\frac{4}{3}, -\frac{4}{3}\right)\).
Summary:
- The roots are \(x = -\frac{2}{3}\) and \(x = -2\).
- The expression factors to \((3x + 2)(x + 2)\).
- The vertex is at \(\left(-\frac{4}{3}, -\frac{4}{3}\right)\).
Feel free to ask if you have more specific questions or need further explanations about any of these steps!
2 answers