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Chapter 11: Problem 30

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. $$ x^{2}-8 x+ $$ _____

Short Answer

Expert verified
Add 16; the trinomial factors to \((x-4)^2\).

Step by step solution

01

Determine the Proper Constant

To make a trinomial a perfect square, use the formula for a perfect square trinomial: \(a^2 - 2ab + b^2 = (a-b)^2\). Identify \(a\) and \(b\). Here, \(a = x\) and \(-2b = -8\). Solve for \(b\) by using \(-2b = -8\), which gives \(b = 4\). Thus, the proper constant to add is \(b^2 = 4^2 = 16\).
02

Write and Verify the Trinomial

Now with the proper constant, the trinomial becomes \(x^2 - 8x + 16\). Check if this new trinomial fits the form \((a-b)^2\). Using \(a = x\) and \(b = 4\), \((x - 4)^2 = x^2 - 8x + 16\), confirming that this trinomial is a perfect square.
03

Factor the Trinomial

The perfect square trinomial \(x^2 - 8x + 16\) can be factored using its square form. Since we established that \((x-4)^2 = x^2 - 8x + 16\), the factorization of the trinomial is \((x-4)^2\).

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Key Concepts

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

Trinomial Factorization
Trinomial factorization involves breaking down a three-term polynomial into simpler expressions that can be multiplied together to yield the original polynomial. Imagine you have a trinomial, such as in our exercise:
  • The expression starts with two terms, specifically an x squared term and another involving x.
  • To factor it, we must first make it a perfect square trinomial by adding a constant.
Once the trinomial becomes a perfect square trinomial, you can express it as the square of a binomial.
For example, the trinomial:
  • \(x^2 - 8x + 16\)
  • can be written as \((x-4)^2\), meaning it factors into two identical binomials: \((x-4)\) and \((x-4)\).
This factorization makes solving equations and simplifying expressions much easier.
Binomial Addition
Binomial addition can sometimes be necessary for creating a perfect square trinomial. In this context, it means that we are adding the correct number or constant to a binomial to complete it.
An easy way of thinking about binomial addition is to consider it as adding the missing piece that upgrades a previously incomplete puzzle:
  • Initially, you might have an expression like \(x^2 - 8x\).
  • To convert this expression into a perfect square, you need to add a term that completes the square.
The missing number is found by solving for \(b\) in the equation \(-2b = -8\). When you find that \(b = 4\), the number you add is \(b^2 = 4^2 = 16\).
This addition creates the trinomial \(x^2 - 8x + 16\), which can be factored into a perfect square binomial \((x - 4)^2\).
Completing the Square
Completing the square is a versatile algebraic technique that transforms a quadratic expression into a perfect square trinomial.
Here鈥檚 how you can complete the square step-by-step:
  • First, identify the quadratic term and the linear term from the binomial.
  • In our example, the expression \(x^2 - 8x\), the coefficients are such that \(-2b\) equates to \(-8\), solving this gives \(b = 4\).
  • Next, find \(b^2\) and add it to the binomial, thus turning it into a trinomial.
  • This results in the trinomial \(x^2 - 8x + 16\), which is now a perfect square trinomial.
  • Finally, rewrite the trinomial in the squared binomial form. Here, it becomes \((x-4)^2\).
This method is not only invaluable for solving quadratic equations but also simplifies expressions for graphing and deeper analysis.

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Most popular questions from this chapter

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ F(x)=2 x^{2}-5 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ f(x)=(x-6)^{2} $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ h(x)=x^{2}-4 $$

Sketch the graph of each function. See Section 11.5 $$ f(x)=3(x-4)^{2}+1 $$

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. g(x)=-7 x^{2}+x+1
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