Final answer:
The minimum value of the quadratic expression x²+8x+10 is found by completing the square, resulting in the vertex form of the equation where the y-coordinate of the vertex represents the minimum value. The expression completes to (x+4)² - 6, revealing that the minimum value is -6.
Step-by-step explanation:
To find the minimum value of the quadratic expression x²+8x+10, we need to complete the square or use the vertex form of a quadratic equation. The vertex form is y = a(x-h)² + k, where (h, k) is the vertex of the parabola, and the value of k is the minimum value if a > 0.
Let's complete the square for the given expression:
- Group the x-terms together: x² + 8x + 10
- Divide the coefficient of x by 2 and square it: (8/2)² = 16
- Add and subtract this number inside the parentheses: x² + 8x + 16 - 16 + 10
- Rewrite as a perfect square trinomial: (x + 4)² - 6
The minimum value occurs at the vertex, (-4, -6), of the parabola represented by the expression. Therefore, the minimum value of the expression is -6.