Question

Determine what number should be added to complete the square of each expression. Then factor each expression. $$ y^{2}-6 y $$

Short answer

Add 9 to complete the square. The factored form is \( (y - 3)^{2} - 9 \).

Step-by-step solution

- Identify the coefficient of the linear term

The given quadratic expression is \( y^{2} - 6y. \) Identify the coefficient of the linear term, which is -6 in this case.

- Calculate half of the coefficient and square it

Take half of the coefficient of the linear term (which is -6), and then square it: \((-6 / 2)^{2} = (-3)^{2} = 9. \) This is the number that needs to be added to complete the square.

- Add and subtract the number to complete the square

Rewrite the expression by adding and subtracting 9: \( y^{2} - 6y + 9 - 9. \)

- Factor the perfect square trinomial

Group the perfect square trinomial and the constant: \( (y^{2} - 6y + 9) - 9. \) Factor the trinomial to get \( (y - 3)^{2} \), so the expression becomes \( (y - 3)^{2} - 9. \)

Key concepts

Quadratic Expression

A quadratic expression is a polynomial of the form ax^{2} + bx + c, where a, b, and c are constants, and a ≠ 0. In the equation y^{2} - 6y, we see that it’s a quadratic expression because it involves the squared variable y^{2}. The standard quadratic expression has three parts:

• The quadratic term (e.g. y^{2}), which is the term with the highest exponent.
• The linear term (e.g. -6y), which is the term with the variable raised to the power of 1.
• The constant term, which in this case is 0 since it is not explicitly shown.

Understanding these parts helps us in various operations like completing the square and factoring trinomials.

Factoring Trinomials

Factoring trinomials is the process of rewriting a trinomial (polynomial with three terms) as a product of two binomials. Let's consider the quadratic expression y^{2} - 6y + 9. This is a trinomial that can be factored. The key steps to factoring this are:

• Identify two numbers that multiply to give the constant term (9) and add to give the linear coefficient (-6).
• In this case, those numbers are -3 and -3.
• We then rewrite the trinomial as (y - 3)(y - 3), which is the same as (y - 3)^{2}. The expression thus becomes (y - 3)^{2} - 9.

Factoring trinomials makes it easier to work with them, especially in solving quadratic equations.

Perfect Square Trinomial

A perfect square trinomial is a special type of trinomial that can be written as the square of a binomial. It takes the form (ax)^{2} + 2abx + b^{2}, which can be factored into (ax + b)^{2}. For instance, in the expression y^{2} - 6y + 9, we:

• Look for a perfect square trinomial pattern.
• Notice that (y)^{2} is the square of y and 9 is the square of 3.
• The middle term, -6y, fits perfectly because -6y = 2×(-3)×y.
• Thus, y^{2} - 6y + 9 can be written as (y - 3)^{2}.

Recognizing this pattern is crucial for simplifying quadratic expressions and solving equations.

Linear Coefficient

The linear coefficient is the coefficient of the linear term in a quadratic expression. In y^{2} - 6y, the linear coefficient is -6. To complete the square, follow these steps:

• Take half of the linear coefficient and then square it.
• For y^{2} - 6y, half of -6 is -3, and (-3)^{2} gives us 9.
• Add and subtract this value to the expression: y^{2} - 6y + 9 - 9.
• This transforms the expression into (y - 3)^{2} - 9.

The linear coefficient guides us in completing the square, which simplifies solving and factoring quadratic expressions.