Radical and rational equations — Digital SAT Math practice
Solve with radicals and fractions; check for extraneous solutions.
Key terms
- Radical Equation: An equation containing a variable inside a square root or other radical symbol.
- Rational Expression: A fraction where the numerator and denominator are polynomials.
- Extraneous Solution: A value obtained during solving that satisfies the transformed equation but not the original.
- Domain Restriction: Values excluded from possible solutions because they make a denominator zero or create negative radicands.
- Common Denominator: The least multiple shared by all denominators, used to combine or clear fractions.
Common traps
- Forgetting to check solutions after squaring accepts extraneous roots.
- Cancelling individual terms instead of common factors when simplifying a rational expression.
- Assuming sqrt(a + b) equals sqrt(a) + sqrt(b); it does not.
- Ignoring values that make a denominator zero or a radicand negative.
Key ideas to know
Solving Radical Equations
Solve sqrt(2x + 5) = x and identify any extraneous solutions. Show your steps.
Square both sides to get 2x + 5 = x2, rearrange to x2 − 2x − 5 = 0, solve for x = 1 ± sqrt(6). Check each in the original equation; only x = 1 + sqrt(6) works because radicals yield nonnegative results.
Watch out: Squaring both sides can create false solutions that satisfy the squared equation but fail the original radical equation, so always substitute back to verify.
Solving Rational Equations
Solve 3/(x−2) + 4/(x+1) = 5/(x2−x−2). What is the valid solution?
Factor denominator to (x−2)(x+1). Multiply by LCD to get 3(x+1) + 4(x−2) = 5. Solve 7x − 5 = 5, so x = 10/7. Check denominators: neither becomes zero, so x = 10/7 is valid.
Watch out: Multiplying by the LCD removes fractions but may introduce values that originally made a denominator zero; always verify each candidate against the original denominators.
Undefined Rational Expressions
For the expression (x2 − 9)/(x2 − 4x + 3), for which value(s) of x is the expression undefined?
Set the denominator equal to zero and solve. x2 − 4x + 3 = 0 factors to (x−3)(x−1) = 0, so the expression is undefined at x = 1 and x = 3.
Watch out: Students often set the numerator to zero instead of the denominator when looking for undefined points, confusing zeros with vertical asymptotes or holes.
Simplifying Rational Expressions
A student simplifies (x + 5)/(x + 7) by cancelling the x terms to get 5/7. What is wrong with this step?
You cannot cancel common terms that are added or subtracted; you must factor first and cancel only common factors. The expression (x+5)/(x+7) is already in simplest form because there are no common factors to cancel.
Watch out: Cancelling terms that are added or subtracted across a fraction bar violates algebra rules and only works when those terms are multiplied factors.
Radical Multiplication and Simplification
Evaluate sqrt(18) × sqrt(2) + sqrt((−3)2). Show the exact value.
Use sqrt(a)×sqrt(b)=sqrt(ab) to get sqrt(36)=6. Use sqrt(x2)=|x| to get sqrt(9)=3. Add them for a total of 9.
Watch out: Assuming sqrt(x2) always equals x ignores negative inputs; the correct simplification is |x|, which prevents sign errors when x is negative.
Sample questions
Question 1
For x not equal to 3, the expression (x^2 - 9)/(x - 3) is equivalent to:
- x - 3
- x + 3
- x^2 + 3
- 3
Show answer and explanation
Answer: x + 3
x^2 - 9 = (x - 3)(x + 3), so the expression simplifies to x + 3.
Question 2
Solve for x: 2/(x - 1) = 4/(x + 1).
- 1
- -3
- 0
- 3
Show answer and explanation
Answer: 3
Cross-multiply: 2(x + 1) = 4(x - 1), so 2x + 2 = 4x - 4 and x = 3.