- Identify the coefficient \(b\) of the \(x\) term as 4.
- Calculate the value needed to complete the square inside the parentheses as \(\left(\frac{4}{2}\right)^2 = 4\).
- Determine the value to add outside by calculating \(-8 \times 4 = -32\) to keep the expression balanced.
- The completed expression is \(y = 8(x^2 + 4x + \boxed{4}) + 17 + (\boxed{-32})\).
Explanation
- Analyzing the Quadratic Function
The goal is to convert the quadratic function \(y = 8x^2 + 32x + 17\) into vertex form by completing the square. The process begins with the given factored form: \(y = 8(x^2 + 4x) + 17\). We need to determine the values that fill the boxes in the expression \(y = 8(x^2 + 4x + \square) + 17 + \square\) such that the equality holds.
- Completing the Square Inside the Parentheses
To complete the square for the expression \(x^2 + 4x\), we look for a constant term such that \(x^2 + 4x + k\) is a perfect square trinomial of the form \((x + h)^2 = x^2 + 2hx + h^2\). Comparing \(4x\) to \(2hx\), we find \(2h = 4\), which gives \(h = 2\). Therefore, the constant term needed is \(h^2 = 2^2 = 4\). This is the value that fills the first box. Adding 4 inside the parenthesis completes the square, resulting in \(8(x+2)^2\).
- Balancing the Equation and Vertex Form
When we add 4 inside the parentheses, we are effectively adding \(8 \times 4\) to the entire expression because the 4 is multiplied by the leading coefficient of 8. To maintain the equality of the function, we must subtract the same amount from the total. Thus, we subtract \(8 \times 4 = 32\) from the outside of the parentheses. The second box is therefore \(-32\). This keeps the expression balanced: \(y = 8(x^2 + 4x + 4) + 17 - 32\). Simplified, this becomes \(y = 8(x+2)^2 - 15\).
- Final Result
In conclusion, the first box is filled with 4 and the second box is filled with -32. The completed statements are:
\(y = 8(x^2 + 4x + \boxed{4}) + 17 + (\boxed{-32})\).
Examples
Completing the square is a fundamental technique in algebra used to solve quadratic equations, analyze parabolas, or find the minimum or maximum values of a quadratic function. For example, if you are designing a parabolic arch for a bridge, converting the equation to vertex form allows you to easily identify the highest point (the vertex) of the arch, which is crucial for structural calculations and safety. By mastering this method, you gain a powerful tool for understanding geometric shapes and optimizing designs in engineering and physics.