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unit recap: a ___________ number can be turned into a fraction. termina…

1. A $\boldsymbol{\text{rational}}$ number can be turned into a fraction. 2. An $\boldsymbol{\text{irrational}}$ number cannot be turned into a fraction. 3. Steps for Simplifying …

分类: algebra 更新时间: 2026-02-04

题目

unit recap:
a ___________ number can be turned into a fraction.
terminating and repeating decimals are included.
an ___________ number cannot be turned into a fraction.
steps for simplifying a radical:

  1. find the largest _________ _________ that is a factor of the number
  2. write the factor pair under the square root
  3. separate into two square roots
  4. evaluate the _________ _________
  5. write integer as being multiplied to the remaining radical

multiplying and dividing radicals:

  1. multiply or divide the ___________
  2. multiply or the numbers inside the ___________
  3. simplify

adding and subtracting radicals:

  1. simplify all ___________
  2. combine the coefficients with identical numbers under the ___________

rationalizing the denominator means to get rid of the ___________ in the bottom of the fraction.
steps:

  1. ___________ both the numerator and the denominator by the radical in the denominator
  2. simplify

the discriminant tells us how many solutions a quadratic will have. the formula for the discriminant is: _________. if the discriminant is > 0, there are ___ solutions. if the discriminant is < 0, there are ___ solutions. if the discriminant = 0, there is _____ solution.
quadratic formula: use the general form of a quadratic, ( ax^2 + bx + c = 0 ), to substitute into:

解题步骤

  1. Understand the question

    unit recap:
    a ___________ number can be turned into a fraction.
    terminating and repeating decimals are included.
    an ___________ number cannot be turned into a fraction.
    steps for simplifying a radical:

    1. find the largest _________ _________ that is a factor of the number
    2. write the factor pair under the square root
    3. separate into two square roots
    4. evaluate the _________ _________
    5. write integer as being multiplied to the remaining radical

    multiplying and dividing radicals:

    1. multiply or divide the ___________
    2. multiply or the numbers inside the ___________
    3. simplify

    adding and subtracting radicals:

    1. simplify all ___________
    2. combine the coefficients with identical numbers under the ___________

    rationalizing the denominator means to get rid of the ___________ in the bottom of the fraction.
    steps:

    1. ___________ both the numerator and the denominator by the radical in the denominator
    2. simplify

    the discriminant tells us how many solutions a quadratic will have. the formula for the discriminant is: _________. if the discriminant is > 0, there are ___ solutions. if the discriminant is < 0, there are ___ solutions. if the discriminant = 0, there is _____ solution.
    quadratic formula: use the general form of a quadratic, ( ax^2 + bx + c = 0 ), to substitute into:

  2. Brief Explanations

    Fill in each blank with the correct term based on definitions and rules of real numbers, radicals, and quadratic equations:

    1. Rational numbers are defined as numbers that can be written as a fraction of two integers, including terminating and repeating decimals.
    2. Irrational numbers cannot be expressed as a fraction of two integers (they have non-terminating, non-repeating decimal expansions).
    3. To simplify a radical, first find the largest perfect square factor of the radicand, then evaluate the square root of that perfect square.
    4. When multiplying/dividing radicals, first operate on the coefficients (the numbers outside the radical), then operate on the numbers inside the radicals.
    5. For adding/subtracting radicals, first simplify all radicals, then combine coefficients of terms with identical numbers under the radicals.
    6. Rationalizing the denominator means removing the radical from the denominator, done by multiplying the numerator and denominator by the radical in the denominator.
    7. The discriminant of a quadratic $ax^2+bx+c=0$ is $\boldsymbol{b^2-4ac}$:
    • If $b^2-4ac>0$, there are two real distinct solutions.
    • If $b^2-4ac<0$, there are no real (two complex) solutions.
    • If $b^2-4ac=0$, there is one real repeated solution.
    1. The quadratic formula for $ax^2+bx+c=0$ is given by the standard formula.
  3. Final answer
    1. A $\boldsymbol{\text{rational}}$ number can be turned into a fraction.
    2. An $\boldsymbol{\text{irrational}}$ number cannot be turned into a fraction.
    3. Steps for Simplifying a Radical:
    4. Find the largest $\boldsymbol{\text{perfect square}}$ that is a factor of the number
    5. Evaluate the $\boldsymbol{\text{perfect square root}}$
    6. Multiplying and Dividing Radicals:
    7. Multiply or Divide the $\boldsymbol{\text{coefficients}}$
    8. Multiply or the numbers inside the $\boldsymbol{\text{radicals}}$
    9. Adding and Subtracting Radicals:
    10. Simplify all $\boldsymbol{\text{radicals}}$
    11. Combine the coefficients with identical numbers under the $\boldsymbol{\text{radicals}}$
    12. Rationalizing the Denominator means to get rid of the $\boldsymbol{\text{radical}}$ in the bottom of the fraction.
    13. $\boldsymbol{\text{Multiply}}$ both the numerator and the denominator by the radical in the denominator
    14. The formula for the discriminant is: $\boldsymbol{b^2-4ac}$. If the discriminant is > 0, there are $\boldsymbol{\text{two real distinct}}$ solutions. If the discriminant is < 0, there are $\boldsymbol{\text{no real}}$ solutions. If the discriminant = 0, there is $\boldsymbol{\text{one real repeated}}$ solution.
    15. Quadratic Formula: $\boldsymbol{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

答案

Brief Explanations

Fill in each blank with the correct term based on definitions and rules of real numbers, radicals, and quadratic equations:

  1. Rational numbers are defined as numbers that can be written as a fraction of two integers, including terminating and repeating decimals.
  2. Irrational numbers cannot be expressed as a fraction of two integers (they have non-terminating, non-repeating decimal expansions).
  3. To simplify a radical, first find the largest perfect square factor of the radicand, then evaluate the square root of that perfect square.
  4. When multiplying/dividing radicals, first operate on the coefficients (the numbers outside the radical), then operate on the numbers inside the radicals.
  5. For adding/subtracting radicals, first simplify all radicals, then combine coefficients of terms with identical numbers under the radicals.
  6. Rationalizing the denominator means removing the radical from the denominator, done by multiplying the numerator and denominator by the radical in the denominator.
  7. The discriminant of a quadratic $ax^2+bx+c=0$ is $\boldsymbol{b^2-4ac}$:
  • If $b^2-4ac>0$, there are two real distinct solutions.
  • If $b^2-4ac<0$, there are no real (two complex) solutions.
  • If $b^2-4ac=0$, there is one real repeated solution.
  1. The quadratic formula for $ax^2+bx+c=0$ is given by the standard formula.

Answer

  1. A $\boldsymbol{\text{rational}}$ number can be turned into a fraction.
  2. An $\boldsymbol{\text{irrational}}$ number cannot be turned into a fraction.
  3. Steps for Simplifying a Radical:
  4. Find the largest $\boldsymbol{\text{perfect square}}$ that is a factor of the number
  5. Evaluate the $\boldsymbol{\text{perfect square root}}$
  6. Multiplying and Dividing Radicals:
  7. Multiply or Divide the $\boldsymbol{\text{coefficients}}$
  8. Multiply or the numbers inside the $\boldsymbol{\text{radicals}}$
  9. Adding and Subtracting Radicals:
  10. Simplify all $\boldsymbol{\text{radicals}}$
  11. Combine the coefficients with identical numbers under the $\boldsymbol{\text{radicals}}$
  12. Rationalizing the Denominator means to get rid of the $\boldsymbol{\text{radical}}$ in the bottom of the fraction.
  13. $\boldsymbol{\text{Multiply}}$ both the numerator and the denominator by the radical in the denominator
  14. The formula for the discriminant is: $\boldsymbol{b^2-4ac}$. If the discriminant is > 0, there are $\boldsymbol{\text{two real distinct}}$ solutions. If the discriminant is < 0, there are $\boldsymbol{\text{no real}}$ solutions. If the discriminant = 0, there is $\boldsymbol{\text{one real repeated}}$ solution.
  15. Quadratic Formula: $\boldsymbol{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

Question Image

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Question Analysis

Subject mathematics
Sub Subject algebra
Education Level high school
Difficulty unspecified
Question Type text only
Multi Question No
Question Count 1
Analysis Status completed
Analyzed At 2026-02-04T23:20:15

OCR Text

Show OCR extraction 
unit recap:
a ___________ number can be turned into a fraction.
terminating and repeating decimals are included.
an ___________ number cannot be turned into a fraction.
steps for simplifying a radical:
1. find the largest ___________ ___________ that is a factor of the number
2. write the factor pair under the square root
3. separate into two square roots
4. evaluate the ___________ ___________
5. write integer as being multiplied to the remaining radical
multiplying and dividing radicals:
1. multiply or divide the ___________
2. multiply or the numbers inside the ___________
3. simplify
adding and subtracting radicals:
1. simplify all ___________
2. combine the coefficients with identical numbers under the ___________
rationalizing the denominator means to get rid of the ___________ in the bottom of the fraction.
steps:
1. ___________ both the numerator and the denominator by the radical in the denominator
2. simplify
the discriminant tells us how many solutions a quadratic will have. the formula for the discriminant is: ___________. if the discriminant is > 0, there are _______ solutions. if the discriminant is < 0, there are _______ solutions. if the discriminant = 0, there is _______ solution.
quadratic formula: use the general form of a quadratic, ( ax^2 + bx + c = 0 ), to substitute into:

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