SHOW ALL WORK
1 what is the solution of the equation
sqrt(2x+13)-5=x
2 what is the solution of the equation
3(^5sqrt((x+2)^3))+3=27
3 what is the simplest form of the radical expression
3(^3sqrt(2a)-6(^3sqrt(2a)
Thank you :)
If I have the brackets correct... then the answer to 1.
is...
first add 5 to both sides to obtain...
(sqrt 2x+13) = x + 5
then square both sides to undo the square root
2x + 13 = (x + 5)^2
expand the right hand side
2x + 13 = (x + 5)(x+5)
2x + 13 = x^2 + 10x + 25
then take everything to the right hand side
0 = x^2 +10x + 25 - 2x - 13
0 = x^2 +8x +12
now factor the trinomial
0 = (x +2)(x + 6)
so x+2 = 0 and x+ 6 = 0
that is x = -2 and x = -6
Thank you!
1) To find the solution of the equation sqrt(2x+13)-5=x, we can start by isolating the square root term on one side:
sqrt(2x+13) - x = 5.
Now, let's square both sides of the equation to eliminate the square root:
(sqrt(2x+13) - x)^2 = (5)^2.
Expanding the left side of the equation:
(2x + 13 - 2x(sqrt(2x+13))) + x^2 = 25.
Simplifying the equation:
13 - 2x(sqrt(2x+13)) + x^2 = 25.
Rearranging the equation to get a quadratic equation:
x^2 - 2x(sqrt(2x+13)) - 12 = 0.
Now, let's solve this quadratic equation.
We can use the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = -2(sqrt(2x+13)), and c = -12.
x = [-(-2(sqrt(2x+13))) ± √((-2(sqrt(2x+13)))^2 - 4(1)(-12))] / (2(1)).
Simplifying further:
x = [2(sqrt(2x+13)) ± √(4(2x+13) + 48)] / 2.
x = [2(sqrt(2x+13)) ± √(8x + 104)] / 2.
x = sqrt(2x+13) ± sqrt(8x + 104).
We can now solve for x using each of the two cases:
Case 1: x = sqrt(2x+13) + sqrt(8x + 104).
Squaring both sides of the equation to eliminate the square roots:
x^2 = 2x+13 + 2(sqrt(2x+13))(sqrt(8x + 104)) + 8x + 104.
x^2 = 10x + 217 + 2(sqrt(2x+13))(sqrt(8x + 104)).
Now, we need to isolate the square roots and solve for x:
2(sqrt(2x+13))(sqrt(8x + 104)) = x^2 - 10x - 217.
Squaring both sides again:
4(2x+13)(8x + 104) = (x^2 - 10x - 217)^2.
Expanding and simplifying:
64x^2 + 832x + 416 + 832x + 10816 = x^4 - 20x^3 - 433x^2 - 20x^3 + 400x^2 + 8650x + 433x^2 - 8650x - 187585.
Rearranging the equation and combining like terms:
x^4 - 40x^3 + 1001x^2 - 3202x - 19567 = 0.
Now, we have obtained a quartic (4th degree) equation in terms of x. Unfortunately, solving quartic equations analytically can be quite complex, and there is no general formula for finding their solutions. In this case, numerical methods or approximations would be needed to find the value(s) of x.
Case 2: x = sqrt(2x+13) - sqrt(8x + 104).
Following the same steps as in Case 1, we would obtain another quartic equation:
x^4 + 40x^3 + 1001x^2 + 3202x - 19567 = 0.
Again, we would need numerical methods or approximations to find the value(s) of x in this case as well.
2) To find the solution of the equation 3(^5sqrt((x+2)^3))+3=27, we can start by isolating the radical term:
3(^5sqrt((x+2)^3)) = 27 - 3.
Simplifying the right side:
3(^5sqrt((x+2)^3)) = 24.
Now, divide both sides by 3:
^5sqrt((x+2)^3) = 8.
Next, raise both sides to the power of 1/5 to eliminate the 5th root:
[((x+2)^3)^(1/5)] = 8^(1/5).
Simplifying further:
(x+2)^3 = 8.
Now, we can solve for x by taking the cube root of both sides:
(x+2) = 2.
Subtracting 2 from both sides:
x = 2 - 2.
Simplifying:
x = 0.
Therefore, the solution to the equation is x = 0.
3) To simplify the radical expression 3(^3sqrt(2a)-6(^3sqrt(2a)), we can factor out a common term:
3(^3sqrt(2a)) - 6(^3sqrt(2a)).
Both terms have a common factor of (^3sqrt(2a)), so we can factor it out:
(^3sqrt(2a))(3 - 6).
Simplifying the expression further:
(^3sqrt(2a))( -3).
Therefore, the simplest form of the given radical expression is -3(^3sqrt(2a)).
1. To find the solution of the equation sqrt(2x+13) - 5 = x, you need to isolate the variable x by performing the following steps:
Step 1: Add 5 to both sides of the equation to eliminate the -5 term on the left side:
sqrt(2x + 13) - 5 + 5 = x + 5
sqrt(2x + 13) = x + 5
Step 2: Square both sides of the equation to eliminate the square root:
(sqrt(2x + 13))^2 = (x + 5)^2
2x + 13 = x^2 + 10x + 25
Step 3: Rearrange the equation to form a quadratic equation:
x^2 + 10x + 25 - (2x + 13) = 0
x^2 + 10x + 25 - 2x - 13 = 0
x^2 + 8x + 12 = 0
Step 4: Factorize the quadratic equation (x^2 + 8x + 12) to find the solutions:
(x + 6)(x + 2) = 0
Step 5: Apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero:
x + 6 = 0 or x + 2 = 0
Step 6: Solve for x in each equation:
For x + 6 = 0, subtract 6 from both sides:
x = -6
For x + 2 = 0, subtract 2 from both sides:
x = -2
Therefore, the solutions to the equation sqrt(2x+13) - 5 = x are x = -6 and x = -2.
2. To find the solution of the equation 3√((x+2)^3)) + 3 = 27, you need to isolate the variable x by following these steps:
Step 1: Subtract 3 from both sides of the equation to eliminate the constant term on the left side:
3√((x+2)^3)) + 3 - 3 = 27 - 3
3√((x+2)^3)) = 24
Step 2: Divide both sides of the equation by 3 to isolate the cube root:
(3√((x+2)^3))/3 = 24/3
√((x+2)^3) = 8
Step 3: Square both sides of the equation to eliminate the square root:
(√((x+2)^3))^2 = 8^2
(x+2)^3 = 64
Step 4: Take the cube root of both sides to eliminate the exponent:
∛((x+2)^3) = ∛64
x + 2 = 4
Step 5: Solve for x by subtracting 2 from both sides:
x = 4 - 2
x = 2
Therefore, the solution to the equation 3√((x+2)^3)) + 3 = 27 is x = 2.
3. To simplify the radical expression 3√(2a) - 6√(2a), you can combine similar terms by following these steps:
Step 1: Factor out the common factor, which is √(2a):
√(2a)(3 - 6)
Step 2: Simplify the expression inside the parentheses:
√(2a)(-3)
Step 3: Write the simplified radical expression:
-3√(2a)
Therefore, the simplest form of the radical expression 3√(2a) - 6√(2a) is -3√(2a).