Y=2x’2-x-3
It looks like you have a quadratic equation in the form of \( Y = 2x^2 - x - 3 \). If you want to analyze this equation, you might want to do several things like finding the roots, the vertex, and the axis of symmetry. Here's how you can do that:
1. Find the Roots:
To find the roots of the equation, you can set \( Y = 0 \):
\[ 2x^2 - x - 3 = 0 \]
You can apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 2, b = -1, c = -3 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4(2)(-3) = 1 + 24 = 25 \]
Now substitute into the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{25}}{2 \times 2} \] \[ x = \frac{1 \pm 5}{4} \]
This gives us two solutions:
- \( x = \frac{6}{4} = 1.5 \)
- \( x = \frac{-4}{4} = -1 \)
2. Find the Vertex:
The vertex of a quadratic function can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting the values:
\[ x = -\frac{-1}{2(2)} = \frac{1}{4} \]
To find the y-coordinate of the vertex, substitute \( x = \frac{1}{4} \) back into the original equation:
\[ Y = 2\left(\frac{1}{4}\right)^2 - \left(\frac{1}{4}\right) - 3 \]
Calculating this gives:
\[ Y = 2\left(\frac{1}{16}\right) - \frac{1}{4} - 3 \] \[ Y = \frac{2}{16} - \frac{4}{16} - \frac{48}{16} \] \[ Y = \frac{2 - 4 - 48}{16} = \frac{-50}{16} = -\frac{25}{8} \]
3. Determine the Axis of Symmetry:
The axis of symmetry formula is \( x = -\frac{b}{2a} \). We already calculated this as:
\[ x = \frac{1}{4} \]
Summary:
- The roots of the equation are \( x = 1.5 \) and \( x = -1 \).
- The vertex of the parabola is at \( \left( \frac{1}{4}, -\frac{25}{8} \right) \).
- The axis of symmetry is \( x = \frac{1}{4} \).
If you need more specific information or further analysis, feel free to ask!
