Solve the equation .
Solve the equation by following steps:
step0: Solve for :
step1: Find the domain:
step2: Multiply both sides of the equation by LCD:
step3: Simplify the equation:
step4: Move the expression to the left side:
step5: Rewrite in standard form:
step6: Solve using the quadratic formula:
step7: Simplify the expression:
step8: Separate into possible cases:
step9: Check if the solution is in the defined range:
step10: Find the intersection:
step11: Rewrite:
Solve the equation .
Solve the equation by following steps:
step0: Solve for :
step1: Find the domain:
step2: Multiply both sides of the equation by LCD:
step3: Simplify the equation:
step4: Move the expression to the left side:
step5: Rewrite in standard form:
step6: Solve using the quadratic formula:
step7: Simplify the expression:
step8: Separate into possible cases:
step9: Check if the solution is in the defined range:
step10: Find the intersection:
step11: Rewrite:
The solutions to the equations are:
i) has solutions and .
ii) has solutions and .
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Mind Expander
To solve the first equation, , you can multiply both sides by (assuming ), leading to . Use the quadratic formula to find the roots, and you’ll discover two distinct values for that satisfy this equation.
For the second equation, , similarly multiply by to get . Again, apply the quadratic formula, and you’ll uncover another pair of solutions. Both these equations showcase how quadratic relationships emerge from basic algebraic expressions!