1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Llana [10]
3 years ago
9

3(x – 9)2 + 15 = 195 Solving quadratics with square root

Mathematics
1 answer:
hoa [83]3 years ago
6 0

Answer:

x -  \frac{6 -  \sqrt{80} }{2}  = 3 - \sqrt[2]{5} = 1.472 \\ x -  \frac{6 +  \sqrt{80} }{2}  = 3 -  \sqrt[2]{5} = 7.472

Step-by-step explanation:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

{(3x - 9)}^{2} \ + 15 - (195) = 0  

Step 1 :Evaluate :  (3x-9)2   =  9x^2-54x+81   =   3 • (3x^2 - 18x - 43)    

Step 2: Pull out like factors :

{9x}^{2} - 54x - 99 = 9•( {x}^{2} - 6x - 11)

Step 3: Trying to factor by splitting the middle term

   Factoring   {x}^{2} - 6x - 11

The first term is,  {x}^{2}  its coefficient is  1 .

The middle term is,  { - 6x} its coefficient is -6 .

The last term, "the constant", is - 11

 Step 4: Multiply the coefficient of the first term by the constant  

1 \: • -11 = -11

Step-5 : Find two factors of -11  whose sum equals the coefficient of the middle term, which is -6. 

- 11 + 1 =  - 10 \\  - 1 + 11 = 10

9 • ( {x}^{2}  - 6x - 11)  = 0

Step 6: 

  Solve :   

9 = 0

This equation has no solution.

A a non-zero constant never equals zero.

Find the Vertex of y =  {x}^{2} -6x-11

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 1 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,A{x}^{2}+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the x

coordinate is  3.0000 

Plugging into the parabola formula   3.0000  for  x  we can calculate the  

y -coordinate :   y = 1.0 * 3.00 * 3.00 - 6.0 * 3.00 - 11.0 or   y

= -20.000

Root plot for y =  {x}^{2} - 6x - 11

Axis of Symmetry (dashed)  {x}={ 3.00} 

Vertex at  {x,y} = { 3.00,-20.00} 

 x -Intercepts (Roots) :

Root 1 at  {x,y} = {-1.47, 0.00} 

Root 2 at  {x,y} = { 7.47, 0.00}

(Please click above graph)

Solve

{x}^{2} \times - 6x = 11

 by Completing The Square .

 Add  11  to both side of the equation :

  {x}^{2} -6x = 11

Now the clever bit: Take the coefficient of  x , which is  6 , divide by two, giving  3 , and finally square it giving  9 

Add  9  to both sides of the equation :

  On the right hand side we have :

   11  +  9    or,  (11/1)+(9/1) 

  The common denominator of the two fractions is  1   Adding  (11/1)+(9/1)  gives  20/1 

  So adding to both sides we finally get :

   x2-6x+9 = 20

Adding  9  has completed the left hand side into a perfect square :

{x}^{2} - 6x + 9 \\ (x - 3) • (x - 3)  \\  {(x - 3)}^{2}

Things which are equal to the same thing are also equal to one another. Since

{x}^{2} - 6x + 9 = 20 \: and \:  {x}^{2} - 6x + 9 =  {(x - 3)}^{2}

then, according to the law of transitivity,

  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

= (x - 3)^{2}  \:is \:  \\  = {(x - 3)}^{ \frac{2}{2}}  \\ =   {(x - 3)}^{1}  \\  = (x - 3)

Now, applying the Square Root Principle to  Eq. #3.3.1  we get:

x - 3 =  \sqrt{20}

Add  3  to both sides to obtain:

  

x = 3 +  \sqrt{20}

Since a square root has two values, one positive and the other negative

{x}^{2} - 6x - 11 = 0

   has two solutions:

x = 3 +  \sqrt{20}  \\  or \: x  = 3 -  \sqrt{20}

Solving   

{x}^{2} - 6x - 11

by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  A{x}^{2}+Bx+C= 0, where  A, B  and  C  are numbers, often called coefficients, is given by :

x  =  \frac{ - B± \sqrt{ {B}^{2 - 4AC} } }{2A}

In our case,  A   =     1

                      B   =    -6

                      C   =  -11

Accordingly,  B2  -  4AC   

 = 36 - (-44)

               =  80

Applying the quadratic formula :

x =  \frac{6± \sqrt{80} }{2}

Can  \sqrt{80}

be simplified ?

Yes! The prime factorization of  80   is

   2•2•2•2•5 

To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 80   =  √ 2•2•2•2•5  

 = 2•2•√ 5 

           = ±  4 • √ 5

\sqrt{80}  =  \sqrt{2•2•2•2•5}  \\ =  \sqrt[2•2•]{5}  \\    = \sqrt[± 4•]{5}

\sqrt{5} rounded to 4 decimal digits, is   2.2361

 So now we are looking at:

           x  =  ( 6 ± 4 •  2.236 ) / 2

x  = \frac{  (6 ± 4 •  2.236)}{2}

Two real solutions:

x = \frac{(6+√80)}{2} =3+2 \sqrt{5} = 7.472     or   x = \frac{6 -  \sqrt{80} }{2} =3-2\sqrt{5}  = -1.472

You might be interested in
A teacher had 21 red pens. If the ratio of red pens to blue pens she owned was 3:1, how many pens did she have total? (just need
Ksivusya [100]
Ok so you do 3x7= 21 which you already know. So then you do 1x7=7. Last, you just add 21+7=28 pens in total
7 0
3 years ago
6(-8)=(-8)6 what property is this
vodomira [7]
Commutitive Property of Multiplication



Hope I'm the brainliest!
8 0
3 years ago
Read 2 more answers
Which expression is equivalent to<br> ва ?<br> o areks<br> 22<br> зев<br> обек<br> 22<br> 34
anyanavicka [17]

Answer:

22

Step-by-step explanation:

it is epuivalet

5 0
3 years ago
Read 2 more answers
The choir is selling candy bars to raise money for a trip they sell each candy bar for $5 there are 7 candy bars in each box and
Reika [66]

Answer:$2800

Step-by-step explanation: Lets first figure out how much money a box of candy bars is worth. We know that each candy bar is worth $5 and that there are 7 candy bars per box. We can multiply 7*5 to figure out that each box is worth $35.

Now lets figure out how many boxes can be sold each day. Each student can sell two boxes a day. 5 students selling 2 boxes each is 5*2= 10 boxes total sold a day.

Now finally we use what we found to figure out the amount of money made per day. We know that there are 10 boxes being sold and that each box makes them $35. We just multiply 10 boxes by $35 to get $350 dollars each day.

Since they sell for 8 days we just multiply 8*$350 to get $2800.

8 0
3 years ago
What is the square root of 144
dlinn [17]

Answer:

12

Step-by-step explanation:

12x12 is 144

3 0
3 years ago
Other questions:
  • To which subsets of real numbers does the number -22 belong? Choose all that apply.
    14·1 answer
  • Tom read a novel for 1 1/4 hours and comic books for 1 2/5 hours. <br><br> How long did Tom read?
    9·1 answer
  • Susan watched the game from 1p.m to 1:30.p.m. describe the turn the mintue hand made
    13·1 answer
  • Find the value of n . 80×π×n=1100000​
    10·1 answer
  • I don’t get this problem:// I’m so confused. I need help asap!!!
    8·1 answer
  • 5^3 x 15^20+ 3^9 =<br><br> Mates only
    13·2 answers
  • Which point is (3,0)?
    8·2 answers
  • Solve the system of equations.<br> 2x-6y=8<br> x-3y=9<br><br> Plz help I’m superrr confused
    13·1 answer
  • 5,000,000 + 800,000 + 50,000 + 300<br> In standard form
    15·2 answers
  • Use the coordinate plane to plot the point (3,6). Then
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!