I have to solve each equation using a u substitution and need help understanding how to do this.
The first equation is:
1. x^2/3-2x^1/3-15=0
2. x+4=7 Squareroot x+4
x^2/3-2x^1/3-15=0
(x^(1/3))^2 - 2x^(1/3) - 15 = 0
suppose we let x^(1/3) = y, then we get
y^2 - 2y - 15 = 0
(y-5)(x+3) = 0
y = 5 or y = -3
then x^(1/3)= 5 , cube both sides
x = 125
or
x^(1/3) = -3
x = -27
2.
x+4 = 7√(x+4)
square both sides
x^2 + 8x + 16 = 49(x+4)
x^2 + 8x + 16 = 49x +196
x^2 - 41x - 180=0
(x-45)(x+4) = 0
x = 45 or x = -4
BUT, since we squared, we must verify all answers
if x = 45
LS = 45+4 = 49
RS = 7√49 = 49 , works
if x = -4
LS = 0
RS = 7√0 = 0 , works
so x = 45 or x = -4
Thank you so much this makes the examples provided make much more scense the way you broke them down! Thanks!
To solve the given equations using a u-substitution, we'll follow these general steps:
Step 1: Choose a substitution
Step 2: Compute the derivative
Step 3: Rewrite the equation in terms of the new variable
Step 4: Solve the resulting equation for the new variable
Step 5: Substitute the value of the new variable back into the original equation
Step 6: Solve for x
Let's apply these steps to the given equations:
1. x^(2/3) - 2x^(1/3) - 15 = 0
Step 1: Choose a substitution
In this case, let's choose u = x^(1/3), so u^2 = (x^(1/3))^2 = x^(2/3).
Step 2: Compute the derivative
Differentiate u with respect to x: du/dx = (1/3)x^(-2/3)
Step 3: Rewrite the equation in terms of the new variable
Now, express the original equation in terms of u:
u^2 - 2u - 15 = 0
Step 4: Solve the resulting equation for the new variable
To solve this quadratic equation, we can factor it as follows:
(u - 5)(u + 3) = 0
Setting each factor equal to zero, we get:
u - 5 = 0 => u = 5
u + 3 = 0 => u = -3
Step 5: Substitute the value of the new variable back into the original equation
Substituting the value of u back into the equation u = x^(1/3):
For u = 5:
5 = x^(1/3)
For u = -3:
-3 = x^(1/3)
Step 6: Solve for x
To find x, we need to raise both sides of the equation to the power of 3:
For u = 5:
5^3 = (x^(1/3))^3
125 = x
For u = -3:
(-3)^3 = (x^(1/3))^3
-27 = x
So, the solutions for the equation x^(2/3) - 2x^(1/3) - 15 = 0 are x = 125 and x = -27.
Now, let's move on to the second equation:
2. x + 4 = 7
Step 1: Choose a substitution
In this case, we don't need a u-substitution since the equation is already in a form that can be easily solved.
Step 2: Solve for x
To solve this equation, we can subtract 4 from both sides:
x = 7 - 4
x = 3
Now, let's solve for the square root of x + 4:
Square root of x + 4:
Step 1: Choose a substitution
In this case, let's choose u = x + 4.
Step 2: Compute the derivative
Differentiate u with respect to x: du/dx = 1
Step 3: Rewrite the equation in terms of the new variable
Now, express the original equation in terms of u (since u = x + 4):
sqrt(u) = ?
Step 4: Solve the resulting equation for the new variable
The equation is already in a simplified form, so we don't need to further solve it.
Step 5: Substitute the value of the new variable back into the original equation
Substitute u = x + 4 back into the equation: sqrt(x + 4) = ?
Step 6: Solve for x
To solve for x, we'll square both sides of the equation:
(x + 4) = ?
Now we have an equation, x + 4 = ?, which we need more information to solve. If you provide the missing information, I can guide you further to find the solution.