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2 years ago

10

Given the equation square root of quantity 2x plus 8 end quantity equals 6, solve for x and identify if it is an extraneous solu

tion.

2 answers:

irina [24]2 years ago

5 0

Answer:

No extraneous solution

Step-by-step explanation:

The given equation is

$\sqrt{2x+8}=6$

Taking square on both sides.

$(\sqrt{2x+8})^2=(6)^2$

$2x+8=36$

Subtract 8 from both sides.

$2x+8-8=36-8$

$2x=28$

Divide both sides by 2.

$x=14$

The solution of given equation is 14.

The solutions of an equation are known as extraneous solutions if they are invalid.

Substitute x=14 in the given equation.

$\sqrt{2(14)+8}=6$

$\sqrt{36}=6$

$6=6$

LHS=RHS, so x=14 is a valid solution.

Therefore, the given equation have no extraneous solution.

Alex_Xolod [135]2 years ago

3 0

Answer:

Solution: x = 6

Step-by-step explanation:

Given equation is:

$\sqrt{2x+8}=6$

In order to solve the equation both sides will be squared

$(\sqrt{2x+8})^{2} = (6)^2 \\2x+8 = 36\\2x = 36-8\\2x = 28\\x = 14$

Verifying the solution

$\sqrt{2(14)+8} = 6\\ \sqrt{28+8} = 6\\\sqrt{36} = 6\\6 = 6$

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Find the integral using substitution or a formula.

Nadusha1986 [10]

$\rm \int \dfrac{x^2+7}{x^2+2x+5}~dx$

Derivative of the denominator:
$\rm (x^2+2x+5)'=2x+2$

Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.

$\rm \int \dfrac{x^2+2+5}{x^2+2x+5}~dx$

and then split the fraction,

$\rm \int \dfrac{x^2+2}{x^2+2x+5}~dx+\int\dfrac{5}{x^2+2x+5}~dx$

Based on our previous test, we know that a simple substitution will work for the first integral: $\rm \quad u=x^2+2x+5\qquad\to\qquad du=2x+2~dx$

So the first integral changes,

$\rm \int \dfrac{1}{u}~du+\int\dfrac{5}{x^2+2x+5}~dx$

integrating to a log,

$\rm ln|x^2+2x+5|+\int\dfrac{5}{x^2+2x+5}~dx$

Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.

$\rm x^2+2x+5=(x^2+2x+1)+4=(x+1)^2+2^2$

So we have this going on,

$\rm ln|x^2+2x+5|+\int\dfrac{5}{(x+1)^2+2^2}~dx$

Let's factor the 5 out of the intergral,
and the 4 from the denominator,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\frac{(x+1)^2}{2^2}+1}~dx$

Bringing all that stuff together as a single square,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(\dfrac{x+1}{2}\right)^2+1}~dx$

Making the substitution: $\rm \quad u=\dfrac{x+1}{2}\qquad\to\qquad 2du=dx$

giving us,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(u\right)^2+1}~2du$

simplying a lil bit,

$\rm ln|x^2+2x+5|+\frac52\int\dfrac{1}{u^2+1}~du$

and hopefully from this point you recognize your arctangent integral,

$\rm ln|x^2+2x+5|+\frac52arctan(u)$

undo your substitution as a final step,
and include a constant of integration,

$\rm ln|x^2+2x+5|+\frac52arctan\left(\frac{x+1}{2}\right)+c$

Hope that helps!
Lemme know if any steps were too confusing.

8 0

3 years ago

Carmen simplifies the expression (4 y + 8 x + 6) + 25 + (4 x + 6 y + 7). The coefficient of the variable y in her simplified ans

Gnoma [55]

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5 0

2 years ago

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755,082 rounded to the nearest 10,000

Tems11 [23]

760,000 is the answer

4 0

3 years ago

Read 2 more answers

What is 101,949 rounded to the nearest hundred thousand?

kobusy [5.1K]

Well the answer is 103,000 because when you round it the 1 becomes a 2

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2 years ago

Read 2 more answers

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 65 inches and standard deviation 4 inch

Serhud [2]

Answer:

Step-by-step explanation:

<u>a)</u>

Given that ; X ~ N ( µ = 65 , σ = 4 )

P ( 64 < X < 66 )

From application of normal distribution ;

Z = ( X - µ ) / σ, Z = ( 64 - 65 ) / 4, Z = -0.25

Z = ( 66 - 65 ) / 4, Z = 0.25

Hence, P ( -0.25 < Z < 0.25 ) = P ( 64 < X < 66 ) = P ( Z < 0.25 ) - P ( Z < -0.25 ) P ( 64 < X < 66 ) = 0.5987 - 0.4013

P ( 64 < X < 66 ) = 0.1974

b) X ~ N ( µ = 65 , σ = 4 )

P ( 64 < X < 66 )

From normal distribution application ;

Z = ( X - µ ) / ( σ / √(n)), plugging in the values,

Z = ( 64 - 65 ) / ( 4 / √(12)) = Z = -0.866

Z = ( 66 - 65 ) / ( 4 / √(12)) = Z = 0.866

P ( -0.87 < Z < 0.87 )

P ( 64 < X < 66 ) = P ( Z < 0.87 ) - P ( Z < -0.87 )

P ( 64 < X < 66 ) = 0.8068 - 0.1932

P ( 64 < X < 66 ) = 0.6135

c) From the values gotten for (a) and (b), it is indicative that the probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

4 0

3 years ago