/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 Show that whenever there is just... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 73

Show that whenever there is just one solution of \(a x^{2}+b x+c=0,\) that solution is of the form \(-b /(2 a)\).

Short Answer

Expert verified
When there is one solution, it is \( x = \frac{-b}{2a} \).

Step by step solution

01

Recognize the type of equation

This is a quadratic equation in the form of: \( ax^{2} + bx + c = 0 \).
02

Examine the conditions for a single solution

For a quadratic equation to have exactly one solution, the discriminant must be zero. The discriminant \(\text{\( D \)}\) is given by: \[ D = b^{2} - 4ac \]. For there to be just one solution, set \( D \) equal to zero: \( b^{2} - 4ac = 0 \).
03

Solve for the derived condition

Solve the equation from Step 2: \( b^{2} = 4ac \).
04

Determine the single solution

With \( b^{2} = 4ac \), substitute this result into the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]. Using \( b^{2} = 4ac \), the formula simplifies to: \( x = \frac{-b \pm \sqrt{0}}{2a} \) Since \( \sqrt{0} = 0 \), the equation simplifies to: \[ x = \frac{-b}{2a} \].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Discriminant
The discriminant is a key part of solving quadratic equations. It is denoted by the symbol \( D \) and is found using the formula \[ D = b^{2} - 4ac \]. This value gives us important information about the nature of the solutions of a quadratic equation. When solving quadratic equations, keep the following in mind:
  • If \( D > 0 \), the quadratic equation has two distinct real solutions.
  • If \( D = 0 \), the equation has one real solution (a double root). This means the parabola touches the x-axis at exactly one point.
  • If \( D < 0 \), the equation has two complex solutions (no real solutions), meaning the parabola does not touch the x-axis at all.
For a quadratic equation to have exactly one solution, the discriminant must be zero. This is an important condition which we use to find specific solutions in quadratic equations.
Quadratic Formula
The quadratic formula is a universal tool to solve any quadratic equation of the form \( ax^{2} + bx + c = 0 \). It is given by: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]. This formula uses the discriminant inside the square root to determine the nature of the solutions. Here's how we use the quadratic formula:
  • Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
  • Calculate the discriminant \( D = b^{2} - 4ac \).
  • Plug \( a \), \( b \), and \( D \) into the quadratic formula.

For instance, if \( D = 0 \), the square root term (\( \sqrt{D} \)) will be zero, leaving us with one solution: \[ x = \frac{-b \pm 0}{2a} = \frac{-b}{2a} \]. This scenario confirms the result we need to prove.
Solving Quadratic Equations
To solve a quadratic equation, follow these steps:
  • First, ensure the equation is in the standard form \( ax^{2} + bx + c = 0 \).
  • Calculate the discriminant \( D \).
  • Based on the value of \( D \), determine the nature and number of solutions:
  • If \( D > 0 \), use both \( \pm \sqrt{D} \) in the quadratic formula to find two distinct solutions.
  • If \( D = 0 \), substitute \( D \) into the quadratic formula to get a single solution: \[ x = \frac{-b}{2a} \].
  • If \( D < 0 \), the solutions will involve complex numbers.

In the specific case provided, we have shown that when \( D = 0 \), the quadratic equation \( ax^{2} + bx + c = 0 \) has exactly one solution given by, \[ x = \frac{-b}{2a} \]. This highlights the importance of understanding the discriminant and the quadratic formula in solving quadratic equations efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(g(x)=x^{2}+14 x+49 .\) Find \(x\) such that \(g(x)=49\).

Solve. $$ x^{2}-10 x+25=64 $$

Solve by completing the square. Show your work. $$ x^{2}+6 x=7 $$

Before graphing a quadratic function, Cassandra always plots five points. First, she calculates and plots the coordinates of the vertex. Then she plots four more points after calculating two more ordered pairs. How is this possible?

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}+6 x+7 $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.