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Chapter 1: Problem 13

Given \(a x^{2}+b x+c=0(a \neq 0)\), write the quadratic formula.

Short Answer

Expert verified
The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step by step solution

01

- Understand the Quadratic Equation

The general form of a quadratic equation is given by: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
02

- Recall the Quadratic Formula

The quadratic formula is used to find the roots of the quadratic equation. The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
03

- Apply the Formula

To solve the quadratic equation \(ax^2 + bx + c = 0\), substitute the values of \(a\), \(b\), and \(c\) from the given equation into the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

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Key Concepts

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

quadratic equation
A quadratic equation is a type of polynomial equation of degree 2. It is generally written in the form \br\br \br\br \( ax^2 + bx + c = 0 \). In this form, the highest power of the variable \( x \) is 2, making it a quadratic. The coefficients \( a \), \( b \), and \( c \) are constants, where \( a eq 0 \). If \( a \) were 0, the equation would be linear, not quadratic.\br\br Quadratic equations can have different types of roots, which are the solutions to the equation. A quadratic equation can have:\br\br
  • Two distinct real roots
  • One real root (when the roots are repeated)
  • No real roots (if the roots are complex)
\br\brUnderstanding the structure of a quadratic equation is crucial for solving it.
solving equations
Solving a quadratic equation involves finding the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). There are several methods to solve quadratic equations:\br\br
  • Factoring
  • Completing the Square
  • Using the Quadratic Formula
\br\brThe Quadratic Formula is the most general method and can be used to solve any quadratic equation. The formula is: \br\br \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \br\br Here, \( b^2 - 4ac \) is called the discriminant, and it determines the nature of the roots. By substituting the values of \( a \), \( b \), and \( c \) into the formula, you can find the roots of any quadratic equation.
roots of quadratic equations
The roots of a quadratic equation are the solutions to the equation \( ax^2 + bx + c = 0 \). Based on the discriminant, \( \Delta = b^2 - 4ac \), the roots can be classified as:

  • Real and distinct: if \( \Delta > 0 \)
  • Real and repeated: if \( \Delta = 0 \)
  • Complex (not real): if \( \Delta < 0 \)


These roots can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For instance, if \( \Delta > 0 \), there will be two different values for \( x \) because of the \( \pm \) sign in the formula. If \( \Delta = 0 \), both values of \( x \) will be the same, and we will have a repeated root. If \( \Delta < 0 \), the values of \( x \) will be complex numbers, which include an imaginary component.
general form of quadratic equation
The general form of a quadratic equation is written as:

\[ ax^2 + bx + c = 0 \]

where \( a eq 0 \). Each term in this equation has a specific role:

  • \( ax^2 \) is the quadratic term, affecting the parabolic shape of the graph
  • \( bx \) is the linear term, influencing the slope of the parabola
  • \( c \) is the constant term, which moves the parabola up or down on the graph


A quadratic equation can be solved for its roots, which are the values of \( x \) that make the equation true. Understanding the general form helps in recognizing and solving these equations efficiently.

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Most popular questions from this chapter

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