Solve the equation .
Solve the equation by following steps:
step0: Solve for :
step1: Subtract the terms:
step2: Divide both sides:
step3: Divide the numbers:
step4: Set the exponents equal:
step5: Move the constant to the right side:
step6: Add the numbers:
Solve the equation .
Solve the equation by following steps:
step0: Solve for :
step1: Add the terms:
step2: Divide both sides:
step3: Divide the numbers:
step4: Rewrite in exponential form:
step5: Set the exponents equal:
Solve the equation .
Solve the equation by following steps:
step0: Solve for :
step1: Simplify:
step2: Divide both sides:
step3: Divide the numbers:
step4: Rewrite in exponential form:
step5: Set the exponents equal:
Solve the equation .
Solve the equation by following steps:
step0: Solve for :
step1: Add the terms:
step2: Divide both sides:
step3: Divide the numbers:
step4: Rewrite in exponential form:
step5: Set the exponents equal:
Solve the equation .
Solve the equation by following steps:
step0: Solve for :
step1: Move the expression to the left side:
step2: Rewrite the expression:
step3: Rewrite the expression:
step4: Use substitution:
step5: Rewrite in standard form:
step6: Solve using the quadratic formula:
step7: Simplify the expression:
step8: Simplify the expression:
step9: Separate into possible cases:
step10: Rewrite the fraction:
step11: Substitute back:
step12: Solve the equation for
step13: Find the union:
The solutions to the given equations are:
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Bonus Knowledge
To solve for in each equation step-by-step:
i. Start with . Rewrite it as , so . Thus, . Therefore, since .
ii. For , rewrite as . Combine like terms: gives . Thus, .
iii. The equation simplifies to . Dividing by 3: leads to since .
iv. In , we rewrite it as . Dividing by 2 gives , which means since .
v. For , note that . This rewrites the equation to . Let ; then, the equation becomes . Rearranging gives . Using the quadratic formula results in two possible values for . This can be further simplified for finding .
So the values of are:
i.
ii.
iii.
iv.
v. Solve that quadratic for values of .