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Chapter 9: Problem 26

Find the \(x\) -intercepts for the graph of each equation. $$ 6 x^{2}+36-30 x=0 $$

Short Answer

Expert verified
The x-intercepts are x = 2 and x = 3.

Step by step solution

01

Rewrite the equation in standard form

The given equation is: 6x^{2} + 36 - 30x = 0 Move all terms to one side of the equation to get it into standard quadratic form \(ax^2 + bx + c = 0\): 6x^2 - 30x + 36 = 0
02

Simplify the equation by factoring out the greatest common factor (GCF)

Notice the terms in the equation all have a common factor. Factor out the greatest common factor (GCF) of 6: 6(x^2 - 5x + 6) = 0
03

Factor the quadratic expression

The expression inside the parentheses is a quadratic trinomial. Factor it further: x^2 - 5x + 6 = (x - 2)(x - 3) Plug back the factors into the equation: 6(x - 2)(x - 3) = 0
04

Apply the Zero Product Property

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Set each factor equal to zero and solve for x: (x - 2) = 0 or (x - 3) = 0
05

Solve each equation

Solve the individual equations for x: x - 2 = 0 x = 2 and x - 3 = 0 x = 3
06

State the x-intercepts

The x-intercepts of the given equation are the values of x that you found. The x-intercepts are: x = 2 and x = 3

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Key Concepts

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

Quadratic Equations
Quadratic equations are polynomial equations of degree 2. They have the general form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(aeq 0\). The term \(ax^2\) is called the quadratic term, \(bx\) the linear term, and \(c\) the constant term.

Quadratic equations can be solved by several methods, including factoring, using the quadratic formula, and completing the square. The solutions to a quadratic equation are also known as its roots or x-intercepts. These are the points where the graph of the quadratic function intersects the x-axis.
Factoring
Factoring is a method used to break down a polynomial into simpler terms (called factors) that can be multiplied to get the original polynomial.

In the case of quadratic equations, the goal is to express the quadratic expression as a product of two binomials. For example, if we have \[ x^2 - 5x + 6 \] we want to find two binomials such that: \[ (x - 2)(x - 3) = x^2 - 5x + 6 \] When the quadratic expression is written as \[ (x - 2)(x - 3) \], we say it has been factored.

Factoring makes it easier to solve the quadratic equation because we can then use the Zero Product Property to find the solutions.
Zero Product Property
The Zero Product Property is a fundamental principle of algebra. It states that if the product of two numbers is zero, then at least one of the numbers must be zero.

Mathematically, it's expressed as: \[ a \times b = 0 \] then either \[ a = 0 \] or \[ b = 0 \] or both.

In the context of solving quadratic equations, once we have factored the quadratic equation into a product of two binomials, we set each binomial equal to zero and solve for x. For example, from \[ (x - 2)(x - 3) = 0 \], we get \[ x - 2 = 0 \] or \[ x - 3 = 0 \], leading to \[ x = 2 \] or \[ x = 3 \]. These values are the x-intercepts of the quadratic function.

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Most popular questions from this chapter

Imagine rolling three regular dice and multiplying all three numbers. a. How many number triples are possible when you roll three dice? b. Without finding the products of every possible roll, describe a way you could determine whether an odd product or an even product is more likely. c. Use your method from Part b to determine whether an even product or an odd product is more likely.

Sports A volleyball league has two divisions. At the end of the season, Team \(\mathrm{A}\) is in first place in Division 1 , followed by Team \(\mathrm{B}\) and Team \(\mathrm{C}\) . Team \(\mathrm{D}\) is in first place in Division 2 , followed by Team \(\mathrm{E}\) and Team \(\mathrm{F}\) . League organizers have structured the playoffs as shown below. a. Model this series six times, keeping a tally of the champion teams. Flip a coin or roll a die to determine the winner of each game. b. Which team won the most championships? c. Determine the probability that each team will win the champ-win. (Hint: Assume both teams in each game are equally likely to win. (Hint: Imagine that a series is played eight times, and determine how many times you can expect each team to win the championship.)

Simplify each expression. $$ \frac{7}{k-2}-\frac{5}{2(k-2)} $$

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A line segment with length 8 and slope \(\frac{1}{4}\) is scaled by a factor of \(3 .\) What are the length and the slope of the new segment?
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