Quadratic Expression Maximum Value Calculation
The problem asks for the maximum value of the given quadratic expression. First, we need to simplify the expression and convert it into the standard quadratic form, $ax^2 + bx + c$. Then, we find the vertex of the parabola represented by this expression, as the maximum (or minimum) value occurs at the vertex.
Simplify the Quadratic Expression
Start by expanding and combining like terms:
Given expression: \(10(x - 1) - 5x^2 + 4(2 - x) + 8\)
- Expand the terms: \(10x - 10 - 5x^2 + 8 - 4x + 8\)
- Group like terms: \(-5x^2 + (10x - 4x) + (-10 + 8 + 8)\)
- Combine terms: \(-5x^2 + 6x + 6\)
The simplified quadratic expression is \(-5x^2 + 6x + 6\).
Find the Vertex Coordinates
For a quadratic expression in the form \(ax^2 + bx + c\), the x-coordinate of the vertex is given by the formula \(x = -\frac{b}{2a}\). In our expression, \(-5x^2 + 6x + 6\):
- \(a = -5\)
- \(b = 6\)
- \(c = 6\)
Calculate the x-coordinate of the vertex:
\(x = -\frac{6}{2(-5)}\) \(x = -\frac{6}{-10}\) \(x = \frac{6}{10}\) \(x = 0.6\)
Calculate the Maximum Value
The maximum value of the quadratic expression occurs at the vertex. Substitute the x-coordinate (\(x = 0.6\)) back into the simplified expression \(-5x^2 + 6x + 6\):
Maximum Value \( = -5(0.6)^2 + 6(0.6) + 6 \) \( = -5(0.36) + 3.6 + 6 \) \( = -1.8 + 3.6 + 6 \) \( = 1.8 + 6 \) \( = 7.8 \)
Since the coefficient 'a' (\(-5\)) is negative, the parabola opens downwards, confirming that the vertex represents the maximum value.
Important Questions from Quadratic equation
What number should be subtracted from x3−4x2−8x+11 to make the number divisible by (x+2)?
- Find the value of K if the quadratic equations $2x^2 + Kx + 8 = 0$ and $3x^2 + 4x + 12 = 0$ have both roots common.
- If sum and product of the roots of a quadratic equation are $(4-3\sqrt{2})$ and -28, respectively, then find the quadratic equation.
- If the quadratic equations $4x^2 + bx + 3 = 0$ and $8x^2 + 4x + c = 0$ have both the roots common, find the values for b and c, respectively.
- Determine the nature of the roots of the quadratic equation $3x^2 + 2x + 5 = 0$.
