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frozen [14]
1 year ago
11

If a = √3-√11 and b = 1 /a, then find a² - b²​

Mathematics
1 answer:
Serga [27]1 year ago
5 0

If b=\frac1a, then by rationalizing the denominator we can rewrite

b = \dfrac1{\sqrt3-\sqrt{11}} \times \dfrac{\sqrt3+\sqrt{11}}{\sqrt3+\sqrt{11}} = \dfrac{\sqrt3+\sqrt{11}}{\left(\sqrt3\right)^2-\left(\sqrt{11}\right)^2} = -\dfrac{\sqrt3+\sqrt{11}}8

Now,

a^2 - b^2 = (a-b) (a+b)

and

a - b = \sqrt3 - \sqrt{11} + \dfrac{\sqrt3 + \sqrt{11}}8 = \dfrac{9\sqrt3 - 7\sqrt{11}}8

a + b = \sqrt3 - \sqrt{11} - \dfrac{\sqrt3 + \sqrt{11}}8 = \dfrac{7\sqrt3 - 9\sqrt{11}}8

\implies a^2 - b^2 = \dfrac{\left(9\sqrt3 - 7\sqrt{11}\right) \left(7\sqrt3 - 9\sqrt{11}\right)}{64} = \boxed{\dfrac{441 - 65\sqrt{33}}{32}}

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Simplify:<br> (4p3 + 6p2 – 7) – (8p? – 7 – 3p)
Kipish [7]

Answer:

4p3 + 6p2 - 8p - 7

Step-by-step explanation:

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "p2"   was replaced by   "p^2".  1 more similar replacement(s).

STEP

1

:

Equation at the end of step 1

 (((4 • (p3)) +  (2•3p2)) -  7) -  8p

STEP

2

:

Equation at the end of step

2

:

 ((22p3 +  (2•3p2)) -  7) -  8p

STEP

3

:

Checking for a perfect cube

3.1    4p3+6p2-8p-7  is not a perfect cube

Trying to factor by pulling out :

3.2      Factoring:  4p3+6p2-8p-7

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -8p-7

Group 2:  4p3+6p2

Pull out from each group separately :

Group 1:   (8p+7) • (-1)

Group 2:   (2p+3) • (2p2)

3.3    Find roots (zeroes) of :       F(p) = 4p3+6p2-8p-7

Polynomial Roots Calculator is a set of methods aimed at finding values of  p  for which   F(p)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  p  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  4  and the Trailing Constant is  -7.

The factor(s) are:

of the Leading Coefficient :  1,2 ,4

of the Trailing Constant :  1 ,7

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        3.00    

     -1       2        -0.50        -2.00    

     -1       4        -0.25        -4.69    

     -7       1        -7.00       -1029.00    

     -7       2        -3.50        -77.00    

     -7       4        -1.75        3.94    

     1       1        1.00        -5.00    

     1       2        0.50        -9.00    

     1       4        0.25        -8.56    

     7       1        7.00        1603.00    

     7       2        3.50        210.00    

     7       4        1.75        18.81    

Final result :

 4p3 + 6p2 - 8p - 7

5 0
2 years ago
20. Jane and Richard each painted about 1/5
Tatiana [17]

Answer:

I can't draw a picture but its possible cuz they painted ABOUT 1/5 of the bird house so Jane could have painted a bit more than 1/5 but it would still be about 1/5

3 0
3 years ago
find the length of each isosceles right triangle when the hypotenuse of the given measure given 5 to the square root of 6
iragen [17]

In isosceles right triangle, legs are equal. Let each leg is of x units . And the given hypotenuse is 5 square 6. We use pythagorean identity here, which is

a^2 + b^2 = c^2

Where a=b=x and c= 5 square root 6

x^2 +x^2 =(5 \sqrt6)^2

2x^2 = 150

x^2 =75

x= 5 \sqrt 3

And that's the length of both the legs . ANd that's the required answer .

8 0
3 years ago
PLEASE HELP ME ON THIS
Whitepunk [10]
F(t)=262(1.6)^t
Initially the tundra has: f(0)=?
t=0→f(0)=262(1.6)^0
f(0)=262(1)
f(0)=262 caribou

Answer:
Initially, the tundra has 262 caribou, and every 1 year, the number of caribou increases by a factor of 1.6


4 0
2 years ago
6 millimeters = yards
sertanlavr [38]
6 millimeters equals 0.00656168 yards.
7 0
3 years ago
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