/*! 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 55 Solve each equation. For equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 55

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-20 x+25=0$$

Short Answer

Expert verified
The solution to the equation is \(x = \frac{5}{2}\), verified graphically by a tangent at this point.

Step by step solution

01

Recognize the Type of Equation

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\). In this case, \(a = 4\), \(b = -20\), and \(c = 25\).
02

Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \[D = (-20)^2 - 4(4)(25) = 400 - 400 = 0.\] Since the discriminant is zero, the equation has exactly one real solution.
03

Solve for the Real Solution

When the discriminant is zero, the quadratic equation has a repeated real root. The solution is given by \(x = \frac{-b}{2a}\). Substitute \(b = -20\) and \(a = 4\) into the formula:\[x = \frac{-(-20)}{2 \times 4} = \frac{20}{8} = \frac{5}{2}.\]Thus, the quadratic equation has one real solution: \(x = \frac{5}{2}.\)
04

Graphical Representation

To support the solution graphically, plot the quadratic function \(y = 4x^2 - 20x + 25\) on a coordinate plane. Since there is one real solution, the graph of the parabola will be tangent to the x-axis at \(x = \frac{5}{2}\). This tangency verifies the one real solution found algebraically.

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
When dealing with quadratic equations of the form \(ax^2 + bx + c = 0\), the discriminant is a handy tool. It's calculated using the formula \(D = b^2 - 4ac\). The value of the discriminant tells us about the number and type of solutions the quadratic equation will have.

  • If \(D > 0\), the equation has two distinct real solutions.
  • If \(D = 0\), there's exactly one real solution, which is a repeated root.
  • If \(D < 0\), the equation does not have real solutions, only complex ones.

This equation \(4x^2 - 20x + 25 = 0\) has a discriminant of zero (D = 0). This tells us that it not only has a real solution, but this solution is a double root, occurring at the same point on the number line.
Graphical Representation
Graphically, a quadratic equation is represented as a parabola on the coordinate plane. The shape and position of this parabola are determined by the coefficients of the quadratic equation.

When dealing with the function \(y = 4x^2 - 20x + 25\), it can be useful to visualize this on a graph to understand real solutions.

For our specific case:
  • The parabola will either cross the x-axis, touch it at a single point, or miss it altogether.

Because the discriminant was zero (D = 0), the parabola touches the x-axis at exactly one point, corresponding to the single real solution. Using this representation, we observe:

  • The parabola opens upwards since the coefficient of \(x^2\) is positive.
  • There is a tangency at the x-axis, precisely at \(x = \frac{5}{2}\).
This graphical check serves as a visual confirmation that there is exactly one real solution, as determined by the algebraic method.
Real Solutions
A real solution for a quadratic equation is a value of \(x\) that satisfies the equation, meaning that when it is plugged into \(ax^2 + bx + c = 0\), the equation holds true. Real solutions are of great interest because they correspond to points where the parabola touches or crosses the x-axis.

In our equation, \(4x^2 - 20x + 25 = 0\), we found only one real solution: \(x = \frac{5}{2}\). Here's what happens:
  • The parabola just grazes the x-axis at \(x = \frac{5}{2}\).
  • This behavior is due to the fact that both sections of the parabola open away from the axis, not crossing it, just touching it at that single point.
This type of interaction specifies the only real solution found due to the quadratic's form and the properties of its coefficients. The real root represents the single point of contact with the x-axis, a fascinating geometric presentation of algebraic theory.

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

Divide. $$\frac{3 x^{4}-2 x^{2}-5}{3 x^{2}-5}$$

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=2 x^{3}-3 x^{2}-17 x+30 ; \quad k=2$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$F=\frac{k M v^{4}}{r} \text { for } v$$

Divide. $$\frac{x^{3}-x^{2}+2 x-3}{x^{2}+3}$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then, use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=5 x^{4}+3 x^{2}+2 x-9$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.