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Chapter 10: Problem 1

Fill in the blanks. The standard form of a _______ equation is \(a x^{2}+b x+c=0\).

Short Answer

Expert verified
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).

Step by step solution

01

Identify the type of equation

The equation given is in the form of \(ax^2 + bx + c = 0\). This is a well-known form in mathematics.
02

Recognize the characteristics

This form of equation typically includes a term with \(x^2\), a linear term \(x\), and a constant \(c\). This is characteristic of a specific type of equation that involves second-degree polynomials.
03

Name the equation type

Equations of the form \(ax^2 + bx + c = 0\) are known as quadratic equations. The term 'quadratic' refers to the highest power of the variable \(x\), which is 2 in this case.

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Key Concepts

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

Standard Form of Quadratic Equation
Understanding the standard form of a quadratic equation is crucial when working with one of the most common types of equations in algebra. The standard form is written as \( ax^2 + bx + c = 0 \). In this expression:
  • \( a \) is the coefficient of \( x^2 \), the quadratic term.
  • \( b \) is the coefficient of \( x \), the linear term.
  • \( c \) is the constant term.
The values of \( a \), \( b \), and \( c \) are usually known as coefficients. For the equation to qualify as a quadratic, \( a \) must not be zero. If \( a = 0 \), the equation becomes linear rather than quadratic. By arranging the terms in this order, it becomes easier to apply various solution methods such as factoring, completing the square, or using the quadratic formula. Making sure the equation is in standard form is an essential first step in any method.
Polynomials
Quadratic equations are a subset of a broader mathematical concept known as polynomials. A polynomial is an expression made up of variables, coefficients, and exponents, combined using addition, subtraction, multiplication, and non-negative integer exponents. Quadratic equations are specifically second-degree polynomials, meaning the highest power of the variable \( x \) is 2.Polynomials can come in various degrees:
  • Constant polynomial: Degree 0 (e.g., \( 5 \)).
  • Linear polynomial: Degree 1 (e.g., \( 3x + 2 \)).
  • Quadratic polynomial: Degree 2 (e.g., \( x^2 + 5x + 6 \)).
  • Cubic polynomial: Degree 3 (e.g., \( x^3 - 2x^2 + x - 1 \)).
Identifying quadratic equations as polynomials helps us understand their behavior and solutions. For example, the solutions to quadratic equations are the **roots** of the polynomial, which are the values of \( x \) that make the polynomial equal to zero.
Second-Degree Equations
Second-degree equations refer to equations where the highest power of the variable is 2, like quadratic equations. These equations often arise in various contexts, including physics, engineering, and statistics, where they model parabolic shapes or describe the motion of objects under uniform acceleration.There are several important characteristics of second-degree equations:
  • The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).
  • The vertex of the parabola represents the maximum or minimum point, which is crucial in optimization problems.
  • Second-degree equations can have zero, one, or two real solutions, as determined by the discriminant \( b^2 - 4ac \).
These features are vital for predicting and understanding how changes in the coefficients affect the solutions and shape of the graphs. Exploring second-degree equations enriches one's understanding of algebraic functions and how they apply to real-world problems.

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Most popular questions from this chapter

Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest hundredth. See Using Your Calculator: Solving Quadratic Equations Graphically. $$ 2 x^{2}-0.5 x-2=0 $$

Fill in the blanks. To complete the square on \(x^{2}+4 x\) shown below, what should be added within the parentheses and what should be added outside the parentheses? $$ f(x)=-5\left(x^{2}+4 x+\square\right)+7+ \square $$

For \(f(x)=-x^{2}+6 x-7,\) the value of \(-\frac{b}{2 a}\) is \(3 .\) Find the \(y\) -coordinate of the vertex of the graph of this function.

Use the discriminant to determine the number and type of solutions for each equation. Do not solve. See Example 1 . $$ 3 x^{2}+10 x-2=0 $$

Solve each inequality. Write the solution set in interval notation and graph it. $$ x^{2}+2 x-8<0 $$
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