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Chapter 10: Problem 59

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. \(m^{2}+18 m\)

Short Answer

Expert verified
(m + 9)^{2}

Step by step solution

01

Identify the coefficient of the linear term

The given quadratic expression is \(m^{2} + 18m\). Identify the coefficient of the linear term. Here, the linear term is 18m, and the coefficient is 18.
02

Divide the coefficient by 2

Take the coefficient of the linear term (which is 18), and divide it by 2: \( \frac{18}{2} = 9 \)
03

Square the result

Now, square the result obtained from dividing by 2: \( 9^{2} = 81 \)
04

Add and subtract the squared term inside the expression

To complete the square, add and subtract the squared result (81) inside the original expression: \( m^{2} + 18m + 81 - 81 \)
05

Group the perfect square trinomial

Rewrite the expression grouping the perfect square trinomial: \( (m^{2} + 18m + 81) - 81 \)
06

Factor the perfect square trinomial

Factor the perfect square trinomial: \( (m + 9)^{2} - 81 \)
07

Final binomial square

Write the result as a binomial squared: \( (m + 9)^{2} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial
A perfect square trinomial is a special kind of quadratic expression where you can factor it into a binomial squared. This means it forms a perfect square when factored. You'll often see it in the form ax^2 + bx + c.To determine if a trinomial is a perfect square, ensure:
  • The first and last terms (ax^2 and c) are perfect squares.
  • The middle term is twice the product of the square roots of the first and last terms.
For example, in the trinomial m^2 + 18m + 81, m^2 and 81 are perfect squares, and 18m is twice the product of 9 and m. This transformation can make solving or factoring equations much simpler.
Factoring
Factoring is the process of breaking down a mathematical expression into simpler components called factors. When you have a perfect square trinomial, you can factor it into a binomial squared.Factoring helps to simplify expressions and solve equations more easily. For example, transform m^2 + 18m + 81 into (m + 9)^2, making it easier to handle the equation. Here are key steps:
  • Identify the perfect square trinomial.
  • Find the binomial that squares back to your original trinomial.
These steps are crucial in simplifying your algebraic expressions.
Quadratic Expressions
Quadratic expressions are algebraic expressions of the form ax^2 + bx + c. These expressions occur frequently in algebra and can be factored or solved in various ways. Completing the square is one method for simplifying quadratic expressions.For example, m^2 + 18m can be approached by adding and subtracting the same value to make it a perfect square trinomial. This method transforms the expression into an easier-to-handle binomial squared. It makes solving equations and graphing parabolas more manageable. Understanding quadratics is key in higher-level algebra.
Binomial Squared
A binomial squared expression fits the form (a + b)^2. This means multiplying a binomial by itself. For example, (m + 9)^2 = m^2 + 18m + 81.The steps to create a binomial squared are:
  • Identify the linear term's coefficient and divide by 2.
  • Square the result to add and subtract.
  • Combine into the binomial squared form.
Using these steps, m^2 + 18m becomes (m + 9)^2. This approach transforms the equation, aiding in simplification and problem-solving.

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