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Chapter 7: Problem 113

Factor each expression. \(u^{2}-12 u+36\)

Short Answer

Expert verified
(u - 6)^{2}

Step by step solution

01

- Identify the quadratic form

The given expression is a quadratic equation of the form: \[u^{2} - 12u + 36\]
02

- Recognize perfect square trinomial

Notice that the quadratic equation can be expressed as a perfect square trinomial. A perfect square trinomial has the form: \[a^{2} - 2ab + b^{2}\].
03

- Find squares and products

Identify the squares and the product terms in the equation: \[u^{2} = (u)^{2}\]\[2ab = 12u \text{ means } ab = 6u \text{ so } b = 6\] \[b^{2} = 6^{2} = 36\]
04

- Write trinomial as a square

Rewrite the equation using the identified terms: \[u^{2} - 12u + 36 = (u - 6)^{2}\]

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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. It can be written in the form a^2 - 2ab + b^2.To create a perfect square trinomial, we need three parts:
  • The square of the first term (a^2 )
  • Twice the product of two terms (2ab )
  • The square of the second term (b^2 )
    • In our example, u^2 - 12u + 36:
    • The first term is u^2 , which is (u)^2.
    • The middle term is -12u, which represents 2ab. Here, we see that 2 * u * 6 = 12u , so b = 6 .
    • Finally, the last term is 36 , which is indeed (6)^2 .
      • So, our perfect square trinomial fits the form as (u - 6)^2 .
Quadratic Equation
A quadratic equation is any equation that can be written in the form ax^2 + bx + c = 0, where:
  • 'a' and 'b' are coefficients
  • 'c' is a constant.
    • Quadratic equations describe parabolas when graphed. They can be factored to find roots or solutions. In the problem we're solving, our quadratic is u^2 - 12u + 36 .To find the roots, we factorize the quadratic equation.
Factoring Expressions
Factoring expressions means breaking down a complicated expression into simpler parts, or factors. This process is useful to solve quadratic equations.
Here's how to factor a quadratic like u^2 - 12u + 36:
  • First, check if it is a perfect square trinomial. As shown, u^2 - 12u + 36 is a perfect square trinomial and fits (u-6)^2 .
  • To factor this, we see it can be rewritten from u^2 - 12u + 36 to (u - 6)(u - 6), which simplifies to (u - 6)^2 .
    • Factoring makes solving quadratic equations easier and provides insight into the equation's properties and solutions.

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Most popular questions from this chapter

Factor completely. \(75 m^{3}+12 m\)

Factor completely. \(m^{4}-81\)

Sue and Alan are planning to put a 15 foot square swimming pool in their backyard. They will surround the pool with a tiled deck, the same width on all sides. If the width of the deck is \(w\), the total area of the pool and deck is given by the trinomial \(4 w^{2}+60 w+225\). Factor the trinomial.

A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall. (a) The binomial \(-16 t^{2}+128\) gives the height of the pumpkin \(t\) seconds after it is dropped. Factor the greatest common factor from this binomial. (b) If the pumpkin is thrown down with initial velocity 32 feet per second, its height after \(t\) seconds is given by the trinomial \(-16 t^{2}-32 t+128\). Completely factor this trinomial.

Factor. \(v^{3}-216\)
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