Solve the Equation: x²/25 - 1 = 0 Step by Step

Quadratic Equations with Fraction Simplification

x225−1=0 \frac{x^2}{25}-1=0

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find X
00:03 Multiply by the denominator to eliminate the fraction
00:14 Simplify what we can
00:18 Isolate X
00:27 Extract the root
00:32 When extracting a root there are always 2 solutions (positive, negative)
00:35 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

x225−1=0 \frac{x^2}{25}-1=0

2

Step-by-step solution

To solve the equation x225−1=0\frac{x^2}{25} - 1 = 0, we'll follow these steps:

  • Step 1: Rewrite the equation to isolate the quadratic term. Add 1 to both sides to get:

x225=1\frac{x^2}{25} = 1

  • Step 2: Eliminate the fraction by multiplying both sides by 25:

x2=25x^2 = 25

  • Step 3: Solve for xx by taking the square root of both sides. Remember to consider both the positive and negative roots:

x=±25x = \pm \sqrt{25}

  • Step 4: Since 25=5\sqrt{25} = 5, the solutions to the equation are:

x=5x = 5 or x=−5x = -5

Therefore, the solution to the problem is x=±5 x = \pm 5 .

3

Final Answer

x=±5 x=\pm5

Key Points to Remember

Essential concepts to master this topic
  • Rule: Multiply both sides to eliminate fractions first
  • Technique: Multiply by denominator: x²/25 becomes x² = 25
  • Check: Substitute x = ±5: (±5)²/25 - 1 = 25/25 - 1 = 0 ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting the negative solution
    Don't solve x² = 25 as just x = 5! This misses half the answer because square roots have both positive and negative values. Always write x = ±√25 = ±5 to get both solutions.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

\( \frac{7}{7\cdot8}=8 \)

FAQ

Everything you need to know about this question

Why do I get two answers from one equation?

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Because when you square both positive and negative numbers, you get the same result! Since both 52=25 5^2 = 25 and (−5)2=25 (-5)^2 = 25 , both values satisfy the original equation.

Should I multiply by 25 first or add 1 first?

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Either way works! You can add 1 first to get x225=1 \frac{x^2}{25} = 1 , then multiply by 25. Or multiply everything by 25 first to get x2−25=0 x^2 - 25 = 0 .

How do I know when to use ± in my answer?

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Use ± whenever you take the square root of both sides! If you have x2=number x^2 = \text{number} , then x=±number x = ±\sqrt{\text{number}} .

What if the number under the square root isn't a perfect square?

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Leave it as a square root! For example, if x2=7 x^2 = 7 , then x=±7 x = ±\sqrt{7} . You don't need to calculate the decimal unless specifically asked.

Can I factor this equation instead?

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Yes! You can rewrite x225−1=0 \frac{x^2}{25} - 1 = 0 as x2−25=0 x^2 - 25 = 0 , then factor: (x−5)(x+5)=0 (x-5)(x+5) = 0 . Both methods give x=±5 x = ±5 !

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