Ask question

Ask question

All categories

3 years ago

9

Help find zeros for 9 and 10

1 answer:

Bingel [31]3 years ago

5 0

<span><span> x4-10x2+9=0</span> </span>Four solutions were found :<span> x = 3 x = -3 x = 1 x = -1</span>

Step by step solution :<span>Step 1 :</span>Skip Ad
<span>Equation at the end of step 1 :</span><span> ((x4) - (2•5x2)) + 9 = 0 </span><span>Step 2 :</span>Trying to factor by splitting the middle term

<span> 2.1 </span>    Factoring <span> x4-10x2+9</span>

The first term is, <span> <span>x4</span> </span> its coefficient is <span> 1 </span>.
The middle term is, <span> <span>-10x2</span> </span> its coefficient is <span> -10 </span>.
The last term, "the constant", is  <span> +9 </span>

Step-1 : Multiply the coefficient of the first term by the constant <span> <span> 1</span> • 9 = 9</span>

Step-2 : Find two factors of   9  whose sum equals the coefficient of the middle term, which is <span> -10 </span>.

<span> -9 + -1 = -10 That's it</span>

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -9  and  -1
                     <span>x4 - 9x2</span> - <span>1x2 - 9</span>

Step-4 : Add up the first 2 terms, pulling out like factors :
                    <span>x2 • (x2-9)</span>
              Add up the last 2 terms, pulling out common factors :
                     1 • <span>(x2-9)</span>
Step-5 : Add up the four terms of step 4 :
                    <span>(x2-1)  •  (x2-9)</span>
             Which is the desired factorization

<span>Trying to factor as a Difference of Squares : </span>

<span> 2.2 </span>     Factoring: <span> x2-1</span>

Theory : A difference of two perfect squares, <span> A2 - B2  </span>can be factored into <span> (A+B) • (A-B)

</span>Proof :<span>  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2 <span>- AB + AB </span>- B2 =
        <span> A2 - B2</span>

</span>Note : <span> <span>AB = BA </span></span>is the commutative property of multiplication.

Note : <span> <span>- AB + AB </span></span>equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1
Check : <span> x2  </span>is the square of  <span> x1 </span>

Factorization is :       (x + 1)  •  (x - 1)

<span>Trying to factor as a Difference of Squares : </span>

<span> 2.3 </span>     Factoring: <span> x2 - 9</span>

Check : 9 is the square of 3
Check : <span> x2  </span>is the square of  <span> x1 </span>

Factorization is :       (x + 3)  •  (x - 3)

<span>Equation at the end of step 2 :</span> (x + 1) • (x - 1) • (x + 3) • (x - 3) = 0 <span>Step 3 :</span>Theory - Roots of a product :

<span> 3.1 </span> A product of several terms equals zero.<span>

</span>When a product of two or more terms equals zero, then at least one of the terms must be zero.<span>

</span>We shall now solve each term = 0 separately<span>

</span>In other words, we are going to solve as many equations as there are terms in the product<span>

</span>Any solution of term = 0 solves product = 0 as well.

<span>Solving a Single Variable Equation : </span>

<span> 3.2 </span>     Solve  :    x+1 = 0<span>

</span>Subtract  1 from both sides of the equation :<span>
</span>                     x = -1

<span>Solving a Single Variable Equation : </span>

<span> 3.3 </span>     Solve  :    x-1 = 0<span>

</span>Add  1 to both sides of the equation :<span>
</span>                     x = 1

<span>Solving a Single Variable Equation : </span>

<span> 3.4 </span>     Solve  :    x+3 = 0<span>

</span>Subtract  3 from both sides of the equation :<span>
</span>                     x = -3

<span>Solving a Single Variable Equation : </span>

<span> 3.5 </span>     Solve  :    x-3 = 0<span>

</span>Add  3 to both sides of the equation :<span>
</span>                     x = 3

Supplement : Solving Quadratic Equation Directly<span>Solving <span> x4-10x2+9</span> = 0 directly </span>

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

<span>Solving a Single Variable Equation : </span>

Equations which are reducible to quadratic :

<span> 4.1 </span>    Solve  <span> x4-10x2+9 = 0</span>

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that <span> w = x2</span> transforms the equation into :
<span> w2-10w+9 = 0</span>

Solving this new equation using the quadratic formula we get two real solutions :
   9.0000  or   1.0000

Now that we know the value(s) of <span> w</span> , we can calculate <span> x</span> since <span> x</span> <span> is  </span><span> √<span> w </span></span>

Doing just this we discover that the solutions of
  <span> x4-10x2+9 = 0</span>
  are either :
  x =√<span> 9.000 </span>= 3.00000 or :
  x =√<span> 9.000 </span>= -3.00000 or :
  x =√<span> 1.000 </span>= 1.00000 or :
  x =√<span> 1.000 </span>= -1.00000

Four solutions were found :<span> x = 3 x = -3 x = 1 x = -1</span>

<span>
Processing ends successfully</span>

You might be interested in

To help vegetables stay fresh longer, Liam’s family maintains storage near their home. The total volume of the root cellar is 95

Helga [31]

To help vegetables stay fresh longer, Volume of storage of 951 cubic feet is equal to 26.929 $m^3$

<u>Solution:</u>

Given that to help vegetables stay fresh longer, Liam’s family maintains storage.

Total volume of root cellar that is storage is 951 cubic feet.

Also given that 1 foot = 0.3048 meters

need to find the volume of storage in cubic meter that is $m^3$

$\begin{array}{l}{\text { As } 1 \text { foot }=0.3048 \mathrm{m}} \\\\ {\Rightarrow 1 \text { foot } \times 1 \text { foot }=0.3048 \mathrm{m} \times 0.3048 \mathrm{m}} \\\\ {\Rightarrow 1 \text { foot } \times 1 \text { foot } \times 1 \text { foot }=0.3048 \mathrm{m} \times 0.3048 \mathrm{m} \times 0.3048 \mathrm{m}} \\\\ {=1 \mathrm{foot}^{3}=0.0283 \mathrm{m}^{3}} \\\\ {\Rightarrow 1 \text { cubic feet }=0.0283 \mathrm{m}^{3}}\end{array}$

$\text { So } 951 \text { cubic feet }=951 \times 0.0283 \mathrm{m}^{3}=26.929 \mathrm{m}^{3}$

Hence volume of storage of 951 cubic feet is equal to 26.929 $m^3$

7 0

3 years ago

Question: which angles are vertical angles?

liraira [26]

6 and 7 are vertical.

5 and 8 are vertical.

7 0

3 years ago

Read 2 more answers

QUESTION 8 The probability of concluding that assignable variation exists when only random variation is present is: (I) the prob

Juliette [100K]

Answer:

I, II, and IV

Step-by-step explanation:

(I) the probability of a Type I error.

(II) known as the alpha risk.

(IV) the sum of probabilities in the two tails of the normal distribution.

3 0

3 years ago

How many units long is a segment whose endpoints are (2,3) and (7,15)?

Ahat [919]

Answer:

13 units

Step-by-step explanation:

Distance formula: $d = \sqrt{(x2-x1)^2 + (y2-y1)^2}$

Simply plug in your variables

$d = \sqrt{(7-2)^2 + (15-3)^2}$

$d = \sqrt{5^2 + 12^2}$

$d = \sqrt{25 + 144}$

$d = \sqrt{169}$

d = 13

5 0

3 years ago

The senate in a certain state is comprised of 51 republicans, 48 democrats, and 1 independent. How many committees can be formed

Sedaia [141]

17 since 51 goes into 3 17 times and 2 goes into 48 24 times and 17 is the largest number that can have 3 republicans and 2 democrats. The andwer would be 17 comittees,

5 0

3 years ago