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Chapter 8: Problem 16

Determine whether the quadratic expression is reducible. $$x^{2}+6 x+9$$

Short Answer

Expert verified
Yes, the quadratic expression \(x^2 + 6x + 9\) is reducible. It can be factorized into \((x+3)^2\).

Step by step solution

01

Identify the Coefficients

The coefficients of the quadratic equation \(x^2 + 6x + 9\) are a=1, b=6, and c=9.
02

Check Discriminant

Calculate the discriminant using the formula: \(b^2 - 4ac\). If it's greater than 0, the expression can be reduced.
03

Calculate Discriminant

Let's calculate the discriminant: \(b^2 - 4ac = 36 - 36 = 0\)
04

Determine Reducibility

Since the discriminant equals to zero, the quadratic expression can be reduced.
05

Factorize Quadratic Expression

Now we need to factorize the expression: \(x^2 + 6x + 9 = (x+3)^2\). This expression is reducible since we successfully factorized it.

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Key Concepts

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

Factorization
Factorization is a method used to break down expressions into products of simpler expressions. For quadratic expressions like \(x^2 + 6x + 9\), factorization can simplify the expression or solve equations that involve it.

Here is a simple approach to factorizing a quadratic expression:
  • First, look at the expression to see if it is a perfect square trinomial. In the given example, \(x^2 + 6x + 9\), it matches the form \((x+a)^2 = x^2 + 2ax + a^2\), where \(a = 3\).
  • Rewriting the quadratic as \((x+3)^2\) allows us to confirm that \(x^2 + 6x + 9\) can be expressed as the square of \(x+3\).
  • This factorization tells us that the expression has a double root at \(x = -3\).
Factorization helps us understand the structure of the quadratic expression and shows us that \(x^2 + 6x + 9\) is reducible.
Discriminant
The discriminant is a useful tool for determining the nature of the roots of a quadratic equation. It is derived from the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

The discriminant is the expression under the square root in the formula: \(b^2 - 4ac\). It helps us decide whether the quadratic can be factorized over real numbers.

The discriminant values lead to different conclusions:
  • If \(b^2 - 4ac > 0\), there are two distinct real roots, and the expression can be factorized.
  • If \(b^2 - 4ac = 0\), there is one real root, and the quadratic is a perfect square (as seen in our example).
  • If \(b^2 - 4ac < 0\), there are no real roots, and the quadratic cannot be factorized over the reals.
The discriminant was \(0\) for the expression \(x^2 + 6x + 9\), indicating the presence of a double root and confirming its reducibility through factorization.
Reducibility
Reducibility refers to the ability to express a quadratic expression as a product of simpler polynomials, often leading to easier evaluation or solution.

To determine the reducibility using factorization:
  • We first compute the discriminant to check if the quadratic has real roots that allow factorization.
  • For the expression \(x^2 + 6x + 9\), the discriminant is zero, suggesting it is reducible and can be factorized into \((x+3)^2\).
  • This means instead of working with the whole quadratic directly, we can think of it in terms of repeated factors. This simplifies understanding and solving quadratic expressions.
Understanding reducibility is crucial because it allows you to predict the behavior of quadratic expressions and their graphs using factorization methods.

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Most popular questions from this chapter

A cab company charges \(\$ 4.50\) for the first mile of a passenger's fare and \(\$ 1.50\) for every mile thereafter. If it is snowing, the fare is increased to \(\$ 5.50\) for the first mile and \(\$ 1.75\) for every mile thereafter. All distances are rounded up to the nearest full mile. Use matrix addition and scalar multiplication to compute the fare for a \(6.8-\) mile trip on both a fair-weather day and a day on which it is snowing.

For the given matrices \(A, B,\) and \(C,\) evaluate the indicated expression. $$\begin{aligned}&A=\left[\begin{array}{rr}3 & -8 \\\2 & 4\end{array}\right] ; \quad B=\left[\begin{array}{rr}-6 & 0 \\\0 & -6\end{array}\right] ; \quad C=\left[\begin{array}{rr}3 & 5 \\\\-2 & 6\end{array}\right]\\\&(A+2 B) C\end{aligned}$$

Involve positive-integer powers of a square matrix \(A . A^{2}\) is defined as the product \(A A ;\) for \(n \geq 3, A^{n}\) is defined as the product \(\left(A^{n-1}\right) A\) Find \(\left(A^{2}\right)^{-1}\) and \(\left(A^{-1}\right)^{2},\) where \(A=\left[\begin{array}{rr}1 & -2 \\ -1 & 3\end{array}\right] .\) What do you observe?

If \(A=\left[\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right]\) and \(B=\left[\begin{array}{cc}2 & 2 a+b \\ b-a & 6\end{array}\right],\) for what values of \(a\) and \(b\) does \(A B=B A ?\)

At a certain gas station, the prices of regular and high-octane gasoline are \(\$ 2.40\) per gallon and \(\$ 2.65\) per gallon, respectively. Use matrix scalar multiplication to compute the cost of 12 gallons of each type of fuel.
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