- Identify the linear term \(8x\) and set it equal to \(2dx\) to find \(d = 4\).
- Square the value of \(d\) to find the constant to add, which is \(4^2 = 16\).
- Form the perfect square trinomial by adding 16 to the original expression to get \(x^2 + 8x + 16\).
- State the final result: \(\boxed{\text{add } 16; (x+4)^2}\).
Explanation
- Understanding the Goal
To transform the binomial expression \(x^2 + 8x\) into a perfect square trinomial, we aim to match it with the standard form of a perfect square trinomial, which is \((x + d)^2 = x^2 + 2dx + d^2\). This process is known as completing the square. By comparing our given expression \(x^2 + 8x\) with the expanded form \(x^2 + 2dx + d^2\), we can identify the value of \(d\).
- Determining the Constant d
By comparing the linear term \(8x\) with the standard form \(2dx\), we set up the equation \(2d = 8\). Dividing both sides of this equation by 2, we find that \(d = 4\). This tells us that the perfect square binomial will be in the form \((x + 4)^2\).
- Completing the Square and Finalizing the Trinomial
Expanding the binomial \((x + 4)^2\), we get \(x^2 + 2(x)(4) + 4^2\), which simplifies to \(x^2 + 8x + 16\). Therefore, to turn the expression \(x^2 + 8x\) into a perfect square, we must add the square of \(d\), which is \(4^2 = 16\). Adding 16 to our initial expression results in the perfect square trinomial \(x^2 + 8x + 16\), which can be written as \((x + 4)^2\).
Examples
Completing the square is a fundamental technique in algebra used to solve quadratic equations or simplify expressions. For example, if you are designing a rectangular garden and you know the area is modeled by the expression \(x^2 + 8x\), completing the square helps you rewrite the area as \((x+4)^2 - 16\). This form makes it much easier to visualize the relationship between the side lengths and the total area, allowing you to quickly determine the dimensions that satisfy specific constraints in landscape architecture or construction planning.