Surd Equations and Exam Problems
Ages 14–18Difficulty adapts to you
Surd Equations and Exam Problems
Treat a surd as an exact number while solving. Collect like surds, isolate a square root before squaring, and check every solution in the original equation because squaring can introduce an answer that was not valid. In geometry and algebra questions, keep exact surd answers until the final line, then give a decimal only if the question requests one.
Your goals
- ✓You can solve simple equations involving surds and present exact answers clearly.
- ✓You can use exact value, approximation, isolate correctly.
- ✓You can isolate a root before squaring and substitute every solution into the original equation.
Words to know
- exact value:
- A value written without rounding, such as 3√2.
- approximation:
- A nearby decimal value, often marked with ≈.
- isolate:
- Get one expression by itself on one side of an equation.
Examples
- 1If x + √3 = 7, subtract √3 to get x = 7 − √3.
- 2If 2x = √5, divide by 2 to get x = √5/2.
- 3For √(x + 1) = 3, square both sides: x + 1 = 9, so x = 8; check √9 = 3.
- 4For a diagonal of a 1-by-2 rectangle, d² = 1² + 2² = 5, so d = √5 units exactly.
- 5If an answer is 3√2, do not write 6 unless you are giving an approximation; 3√2 is the exact value.
More than one way to solve it
Try each method, then choose the one you can explain and check most clearly.
Method 1
Isolate then square
- 1.Move other terms away from the root.
- 2.Square both sides.
- 3.Solve and substitute into the original equation.
Example: √(x+1)=3 gives x+1=9, so x=8; √9 checks the answer.
Method 2
Exact-form discipline
- 1.Keep roots and fractions exact.
- 2.Simplify before approximating.
- 3.Add a decimal only if the question asks for it.
Example: A diagonal of √5 units is exact; 2.236 units is an approximation.
Check these
- • Do not round too early or accept a value created by squaring without checking it.
- • Keep every number, symbol, label, and unit in its correct place.
- • Choose the method that makes each step easiest to explain and check.
Helpful habits
- • Read the whole question before choosing what to do.
- • Show one clear step at a time and check each step.
- • If you are stuck, draw a model, try smaller numbers, or revisit the example.
Now try your own
Create your own surd equations and exam problems example, solve it, and explain how you checked it.
Spark
Creativity & Innovation“Create your own maths challenge and solve it!”