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Given parameters:

  Equation:

          (x-4)²=9

Problem: Solve the equation by both factoring and extracting the square root.

Solution:

    Starting equation:

                  (x-4)²=9

         Subtracting 9 from both sides brings us to zero;

                (x-4)² - 9 = 0

                (x -4)² - 3² = 0

  This fits the concept of the difference of squares;

          x² - y² = (x + y)(x-y)

Let x  = x-4 and y = -3

  Then input and solve;

              (x - 4 -3)(x - 4 -(-3)) = 0

              (x - 7)(x - 1) = 0

S thus,

             x - 7 = 0 or x-1 = 0

             x = 7 or 1

          (x-4)² = 9

         √(x-4)² = √9

              x - 4 = 3

              x  = 4 + 3 = 7; however, this is not the sole solution

Thus, direct extraction of the square root is not the method for complete solutions.

Answer:

" Vertical angles are equal " ⇒ 2nd answer

Step-by-step explanation:

* Refer to the attached illustration

- Three lines intersect at point D.

- We have to identify the missing justification in step 3.

∵ Line FA intersects line EC at point D.

- When two lines cross, the angles created are referred to as

 vertical angles.

- By the vertical angles theorem, vertical angles are equal.

Thus, ∠ADC and ∠FDE are vertical angles.

Since vertical angles are equal

∴ ∠EDF ≅ ∠ADC

Thus, m∠EDF ≅ m∠ADC

Given that m∠EDF = 120°.

∵ m∠ADC is the sum of m∠ADB and m∠BDC.

Therefore, m∠ADB + m∠BDC = 120°.

∵ m∠ADB = (3x)° ⇒ given.

∵ m∠BDC = (2x)° ⇒ given.

Thus, 3x + 2x = 120 ⇒ combine like terms.

Thus, 5x = 120 ⇒ divide both sides by 5.

Thus, x = 24.

Column (1)                                                     Column (2)

m∠EDF = 120°                                               given

m∠ADB = 3 x                                                 given

m∠BDC = 2 x                                                 given

∠EDF and ∠ADC are vertical angles           definition of vertical angles

∠EDF is equal to ∠ADC                           vertical angles are equal

                                                                        equal  

m∠ADC = m∠ADB + m∠BDC                        angle addition principle.

m∠EDF = m∠ADC                                          definition of equality.

m∠EDF = m∠ADB + m∠BDC                         substitution.

120° = 3 x + 2 x                                               substitution.

120 = 5 x                                                         addition.

x = 24                                                           division.

∴ The missing justification is " vertical angles are equal "

- From the reasoning above, ∠ADC and ∠FDE are vertical angles and therefore they are equal according to the vertical angle theorem.

Answer:

Step-by-step explanation:

Step 1

Calculate the slope of the dashed line

The slope between two points is computed with the formula:

We consider

(-3,1) and (0,3)

and substitute into it:

Step 2

Determine the equation of the dashed line in slope-intercept form:

The equation we established is:

---> associated problem

Substituting yields:

Step 3

Establish the inequality corresponding to the graph

Given that it is represented by a dashed line, the area to the left of this line is shaded.

Thus,

Refer to the attached diagram for enhanced understanding of the problem.

Response:

Step-by-step explanation:

Shift the decimal points in both the divisor and the dividend.

Transform the divisor (the number you're dividing by) into a whole number by moving its decimal to the furthest right. Simultaneously, adjust the dividend's decimal (the number being divided) the same number of places to the right.

In the quotient (the result), place a decimal point directly over where the decimal point is now located in the dividend.

Proceed with the division as normal, ensuring proper alignment so the decimal point appears correctly.

Align each digit in the quotient directly over the last digit of the dividend utilized in that step.