8x² + 10x + 3

To analyze the quadratic expression \(8x^2 + 10x + 3\), you can perform several operations, such as factoring, finding roots, or determining its vertex.

1. Factoring: To factor the quadratic, we look for two numbers that multiply to \(a \cdot c = 8 \cdot 3 = 24\) and add to \(b = 10\). However, in this case, finding such numbers doesn't lead to integer factors. Therefore, this quadratic does not factor nicely over the integers.

2. Finding the Roots: You can find the roots using the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] where \(a = 8\), \(b = 10\), and \(c = 3\).

   First, calculate the discriminant (\(D = b^2 - 4ac\)): \[ D = 10^2 - 4 \cdot 8 \cdot 3 = 100 - 96 = 4 \] Since the discriminant is positive, there are two distinct real roots.

   Now, substitute into the quadratic formula: \[ x = \frac{{-10 \pm \sqrt{4}}}{2 \cdot 8} = \frac{{-10 \pm 2}}{16} \] This leads to two roots: \[ x_1 = \frac{{-10 + 2}}{16} = \frac{{-8}}{16} = -\frac{1}{2} \] \[ x_2 = \frac{{-10 - 2}}{16} = \frac{{-12}}{16} = -\frac{3}{4} \]

3. Vertex of the Parabola: The vertex \(x\)-coordinate of a parabola given by \(ax^2 + bx + c\) can be found using the formula: \[ x = -\frac{b}{2a} \] For this expression: \[ x = -\frac{10}{2 \cdot 8} = -\frac{10}{16} = -\frac{5}{8} \] To find the \(y\)-coordinate of the vertex, substitute this \(x\) value back into the original expression: \[ y = 8\left(-\frac{5}{8}\right)^2 + 10\left(-\frac{5}{8}\right) + 3 \] \[ = 8 \cdot \frac{25}{64} - \frac{50}{8} + 3 \] \[ = \frac{200}{64} - \frac{400}{64} + \frac{192}{64} \] \[ = \frac{200 - 400 + 192}{64} = \frac{-8}{64} = -\frac{1}{8} \] Thus, the vertex of the parabola is at \(\left(-\frac{5}{8}, -\frac{1}{8}\right)\).

In summary, the quadratic \(8x^2 + 10x + 3\) has two real roots at \(x = -\frac{1}{2}\) and \(x = -\frac{3}{4}\), and its vertex is located at \(\left(-\frac{5}{8}, -\frac{1}{8}\right)\).