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find a positive value of y which satisfies the equation y square + 1 by y minus1 is equal to 5 by 4
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find a positive value of y which satisfies the equation y square + 1 b...
Solving the Equation
To solve the equation y^2 - 1/y - 1 = 5/4, we need to simplify the left-hand side first.
Step 1: Multiply both sides by 4y to get rid of the denominators.
4y(y^2 - 1/y - 1) = 5
Step 2: Expand the left-hand side by distributing 4y.
4y^3 - 4 - 4y = 5
Step 3: Move all the terms to one side of the equation.
4y^3 - 4y - 9 = 0
Step 4: Factor the left-hand side using the Rational Root Theorem.
The possible rational roots of 4y^3 - 4y - 9 = 0 are ±1, ±3/2, and ±9/4. Checking these roots, we find that y = 3/2 is a root.
Step 5: Divide 4y^3 - 4y - 9 by y - 3/2 to get the other two roots.
Using long division or synthetic division, we get:
4y^3 - 4y - 9 = (y - 3/2)(4y^2 + 6y + 6)
The quadratic factor has no real roots, so the other two roots of the equation are complex conjugates:
y = (-3 ± i√3)/2
Step 6: Check that y = 3/2 satisfies the original equation.
y^2 - 1/y - 1 = 5/4
(3/2)^2 - 1/(3/2) - 1 = 5/4
9/4 - 2/3 - 1 = 5/4
5/4 = 5/4
Therefore, the positive value of y that satisfies the equation is y = 3/2.
To solve the equation y^2 - 1/y - 1 = 5/4, we need to simplify the left-hand side first.
Step 1: Multiply both sides by 4y to get rid of the denominators.
4y(y^2 - 1/y - 1) = 5
Step 2: Expand the left-hand side by distributing 4y.
4y^3 - 4 - 4y = 5
Step 3: Move all the terms to one side of the equation.
4y^3 - 4y - 9 = 0
Step 4: Factor the left-hand side using the Rational Root Theorem.
The possible rational roots of 4y^3 - 4y - 9 = 0 are ±1, ±3/2, and ±9/4. Checking these roots, we find that y = 3/2 is a root.
Step 5: Divide 4y^3 - 4y - 9 by y - 3/2 to get the other two roots.
Using long division or synthetic division, we get:
4y^3 - 4y - 9 = (y - 3/2)(4y^2 + 6y + 6)
The quadratic factor has no real roots, so the other two roots of the equation are complex conjugates:
y = (-3 ± i√3)/2
Step 6: Check that y = 3/2 satisfies the original equation.
y^2 - 1/y - 1 = 5/4
(3/2)^2 - 1/(3/2) - 1 = 5/4
9/4 - 2/3 - 1 = 5/4
5/4 = 5/4
Therefore, the positive value of y that satisfies the equation is y = 3/2.
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find a positive value of y which satisfies the equation y square + 1 b...
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