Question

Determine what number should be added to complete the square of each expression. Then factor each expression. $$ x^{2}+10 x $$

Short answer

Add 25 to complete the square, factor as \((x + 5)^{2}\).

Step-by-step solution

- Identify the coefficient of the linear term

The given expression is \(x^{2}+10x\). The coefficient of the linear term \(x\) is 10.

- Divide the coefficient by 2

Divide the coefficient of the linear term (10) by 2: \[ \frac{10}{2} = 5 \]

- Square the result

Square the result obtained from step 2: \[5^2 = 25\]

- Add and subtract the squared value

Add and subtract the squared value (25) inside the expression to maintain equality: \[x^{2} + 10x + 25 - 25\]

- Form a perfect square trinomial

Combine \(x^{2} + 10x + 25\) to form a perfect square trinomial: \[(x + 5)^{2} - 25\]

- Simplify the expression

Factor the trinomial as a square: \[(x + 5)^{2}\]. Therefore, the factored form is \((x + 5)^2\)

Key concepts

Factoring

Factoring is a process of breaking down a complex expression into simpler components called factors. In algebra, factoring often involves manipulating polynomial expressions. For instance, in the given problem, we are working to simplify the quadratic expression by completing the square and then factoring it. The key here is to convert the expression into a form that can be easily factored or solved, such as transforming it into a perfect square trinomial. This technique helps in solving equations, graphing parabolas, and understanding the underlying properties of algebraic expressions. Remember: the factors are expressions that, when multiplied, give you the original polynomial.

Quadratic Expression

A quadratic expression is a polynomial of degree 2. It has the general form of ax^2 + bx + c, where a, b, and c are constants, and x is the variable. In the exercise, the quadratic expression is given as x^2 + 10x. Here, the coefficient a = 1 (the coefficient of x^2), b = 10 (the coefficient of x), and c = 0 (the constant term). Quadratic expressions are essential in algebra because they often describe parabolic graphs and have properties useful in various applications, including physics and engineering. By learning to complete the square, you can transform any quadratic expression into a more manageable form, aiding in both solving and factoring.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions using various algebraic techniques. In the context of this exercise, we used the method of completing the square, a powerful technique for transforming a quadratic expression into a perfect square trinomial. Here’s a quick recap of the steps:

• Identify the linear term's coefficient.
• Divide the coefficient by 2.
• Square the result.
• Adjust the expression by adding and subtracting the squared value.
• Reform the expression into a perfect square trinomial.
• Factor the trinomial.

This systematic approach simplifies the quadratic expression and helps identify its factors. Consistently practicing algebraic manipulation improves your problem-solving skills and conceptual understanding in algebra and beyond.