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Chapter 0: Problem 52

Write the quadratic equation in general form. $$ 13-3(x+7)^{2}=0 $$

Short Answer

Expert verified
The quadratic equation in general form is \(3x^2 + 42x + 134 = 0\).

Step by step solution

01

Expand the equation

Multiplying out the brackets, we obtain \(13 - 3*(x^2 + 14x + 49) = 0\). Distributing the -3 through the parenthesis gives \(-3x^2 - 42x - 147 + 13 = 0\).
02

Simplifying the equation

Combine constant terms in the equation which results in \(-3x^2 - 42x -134=0\).
03

Rearrange in the general form

The quality equation in general form must have positive \(x^2\). Therefore, we multiply all terms by -1 to obtain \(3x^2 + 42x + 134 = 0\).

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Key Concepts

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

Algebra
Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to express real-world problems in a generalized manner. With algebra, you can solve equations and find unknown values. It's like a language that uses numbers, letters, and symbols to simplify and represent mathematical relationships.

Some basic elements of algebra include:
  • Variables: Symbols like \(x\), \(y\), or \(z\) that represent unknown values.
  • Constants: Numbers that have fixed values.
  • Expressions: Combinations of variables and constants, like \(3x + 5\).
  • Equations: Mathematical statements that assert the equality of two expressions, like \(2x + 3 = 7\).
Understanding algebra is key to solving quadratic equations, which are foundational in mathematics.
Expanding Brackets
Expanding brackets involves removing the parentheses from an expression. This is done by distributing each term inside the bracket with the term outside. It's like unpacking a math problem to see all its parts laid out.

In our example, the expression is \(-3(x + 7)^2\). To expand this:
  • First, expand \((x+7)^2\) to \(x^2 + 14x + 49\).
  • Next, distribute the \(-3\) to each term, resulting in \(-3x^2 - 42x - 147\).
This step is crucial for simplifying and rearranging the equation. Expanding brackets helps us break down complex expressions and makes solving equations easier.
General Form
The general form of a quadratic equation is the standard way of writing it. It is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero.

In the general form:
  • \(a\) represents the coefficient of \(x^2\).
  • \(b\) represents the coefficient of \(x\).
  • \(c\) is the constant term.
For our equation, after the necessary simplification and rearrangement, it is written as \(3x^2 + 42x + 134 = 0\).
This form allows you to efficiently apply methods like factoring, completing the square, or using the quadratic formula to solve the equation.
Polynomial Equations
Polynomial equations are equations that involve sums of powers of unknowns, often expressed as multiple terms. The highest power of the variable denotes the degree of the polynomial. In our case, a quadratic equation is a polynomial of degree 2.

Key points about polynomial equations:
  • Degree: The highest power of the variable. Quadratics have a degree of 2.
  • Terms: Each part of the expression separated by a plus or minus sign.
  • Coefficients: The numerical multipliers of the terms, such as 3 in \(3x^2\).
Polynomial equations can represent a wide variety of numbers and patterns, making them essential in mathematics. They are used in many real-world applications, from physics to economics.

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Most popular questions from this chapter

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

The number of lumens (time rate of flow of light) \(L\) from a fluorescent lamp can be approximated by the model $$ L=-0.294 x^{2}+97.744 x-664.875, \quad 20 \leq x \leq 90 $$ where \(x\) is the wattage of the lamp. (a) Use a graphing utility to graph the function. (b) Use the graph from part (a) to estimate the wattage necessary to obtain 2000 lumens.

The set of ordered pairs \(\\{(-8,-2),(-6,0),(-4,0)\), \((-2,2),(0,4),(2,-2)\\}\) represents a function.

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ g(x)=\frac{4-x}{6} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$
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