/*! 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 14 Show that \(6-\sqrt{20}\) is a s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 14

Show that \(6-\sqrt{20}\) is a solution to the equation \(0=0.5 x^{2}-6 x+8\).

Short Answer

Expert verified
Yes, \(6 - \sqrt{20}\) satisfies the equation.

Step by step solution

01

Substitute the Value

To check whether \(x = 6 - \sqrt{20}\) is a solution to the equation \(0 = 0.5x^{2} - 6x + 8\), we begin by substituting the value \(x = 6 - \sqrt{20}\) into the equation.
02

Calculate \(x^2\)

Calculate \(x^2\) by using the expression \(x = 6 - \sqrt{20}\). So, \((6 - \sqrt{20})^2 = 36 - 2\cdot6\cdot\sqrt{20} + 20\). Simplifying further, we get \(36 + 20 - 12\sqrt{20} = 56 - 12\sqrt{20}\).
03

Substitute \(x^2\) into Equation

Substitute \(x^2\) in the equation \(0 = 0.5x^2 - 6x + 8\). We have to substitute \(56 - 12\sqrt{20}\) for \(x^2\), and \(6 - \sqrt{20}\) for \(x\).
04

Simplify the Equation

Substitute \(x = 6 - \sqrt{20}\) and \(x^2 = 56 - 12\sqrt{20}\) into the equation:\[0.5(56 - 12\sqrt{20}) - 6(6 - \sqrt{20}) + 8 = 0\]Next, simplify this to:\[28 - 6\sqrt{20} - 36 + 6\sqrt{20} + 8 = 0\]Combine the terms:\[28 + 8 - 36 = 0\]This results in:\[0 = 0\], proving the initial assumption.

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.

Quadratic Formula
The quadratic formula is a powerful tool for solving polynomial equations of the form \(ax^2 + bx + c = 0\). It allows us to find the values of \(x\) that satisfy the equation. The formula is given by:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Here, \(a\), \(b\), and \(c\) are coefficients of the quadratic equation. The beauty of this formula lies in its ability to find both real and complex roots by evaluating the discriminant (\(b^2 - 4ac\)).
While our current example does not require the use of the quadratic formula directly, understanding it helps in verifying whether proposed values for \(x\) are valid solutions to the equation.
Solution Verification
Solution verification involves checking whether a particular value satisfies a given equation. In our exercise, we need to verify whether \(x = 6 - \sqrt{20}\) is a solution to the equation \(0 = 0.5x^2 - 6x + 8\).
Here's how you can verify:
  • Substitute the proposed solution into the equation.
  • Simplify the equation to check if both sides of the equation are equal.
For instance:
1. Substitute \(x = 6 - \sqrt{20}\) into the quadratic equation.
2. Calculate \((6 - \sqrt{20})^2 = 56 - 12\sqrt{20}\).
3. Substitute \(x^2\) back into the original equation and simplify.
4. Check if the final result equals zero. If yes, the value is indeed a solution.
Simplifying Expressions
Simplifying an expression means reducing its complexity by performing algebraic operations. In our exercise, simplifying expressions is crucial when verifying the solution.
Follow these steps:
  • First, square the expression if needed. For \((6 - \sqrt{20})^2\), apply the formula \((a - b)^2 = a^2 - 2ab + b^2\).
  • Combine like terms to streamline the equation. In this example, combine numerical terms \(36\) and \(20\) to get \(56\), and simplify \(-12\sqrt{20}\).
  • Continue simplifying until the equation is as simple as possible, helping reveal if it's equal to zero.
Properly simplified expressions make it easier to verify solutions and ensure calculations are both correct and visually straightforward.

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

Is a triangle with side lengths of \(9 \mathrm{~cm}, 16 \mathrm{~cm}\), and \(25 \mathrm{~cm}\) a right triangle? Explain.

Find an exact solution to each equation. (Leave your answers in radical form.) a. \(x^{2}=47\) b. \((x-4)^{2}=28\) c. \((x+2)^{2}-3=11\) d. \(2(x-1)^{2}+4=18\)

For each situation, identify whether inductive or deductive reasoning is used. Then state the hypothesis and conclusion. a. The dinosaur Tyrannosaurus rex had sharp teeth. Animals with sharp teeth eat meat. Meat-eating animals are called carnivores. Therefore, Tyrannosaurus rex was a carnivore. b. Krystal adds \(5+6\) and gets 11 . Then she adds \(13+14\) and gets 27 . Then she adds \(92+93\) and gets 185 . Krystal concludes that the sum of any two consecutive integers is odd. c. Kendra uses the properties of numbers to show that \(2(x-3)=10\) is equivalent to \(x=8\).

For Exercises \(7-14\), plot each set of points on graph paper and connect them to form a polygon. Classify each polygon using the most specific term that describes it. Use deductive reasoning to justify your answers by finding the slopes of the sides of the polygons. $$ (-4,-1),(-2,7),(2,6),(3,3) $$

The equation of line \(\ell\) has the form \(A x+B y=C\). What is the slope of a line a. Perpendicular to line \(\ell\) ? b. Parallel to line \(\ell\) ?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.