The expression 6x² + 7x + 10 is a quadratic equation in standard form, where:
- 6 is the coefficient of the x² term.
- 7 is the coefficient of the x term.
- 10 is the constant term.
To analyze it, we can evaluate its properties:
- Shape: The graph of a quadratic function is a parabola. Since the coefficient of x² (which is 6) is positive, the parabola opens upwards.
- Vertex: The vertex of the parabola can be found using the formula x = -b/(2a), where a is the coefficient of x² (6) and b is the coefficient of x (7). Calculating this gives:
x = -7/(2 * 6) = -7/12
This value of x can be substituted back into the original equation to find the corresponding y-coordinate of the vertex.
- Discriminant: The discriminant of the quadratic equation (calculated as b² – 4ac) helps determine the nature of the roots. For our equation:
The discriminant is (7)² – 4*(6)*(10) = 49 – 240 = -191. A negative discriminant indicates that there are no real roots, meaning the parabola does not intersect the x-axis.
- Y-intercept: The y-intercept occurs when x = 0. Evaluating the expression at this value gives:
y = 6(0)² + 7(0) + 10 = 10, so the y-intercept is at the point (0, 10).
In summary, we can state that the expression 6x² + 7x + 10 is a quadratic polynomial that:
- Opens upwards
- Has a vertex whose x-coordinate is -7/12
- Has no real roots
- Intercepts the y-axis at (0, 10)