Solve the Quadratic Equation: x² - 25 = 0

Quadratic Equations with Difference of Squares

Find the value of the parameter x.

x2−25=0 x^2-25=0

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:06 Let's factor 25 to 5 squared
00:11 Let's use the abbreviated multiplication formulas
00:19 Let's use this formula in our exercise
00:35 Let's find what makes each factor zero
00:39 Let's isolate the unknown, this is one solution, now let's find the second
00:45 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

Find the value of the parameter x.

x2−25=0 x^2-25=0

2

Step-by-step solution

We will factor using the shortened multiplication formulas:

a2−b2=(a−b)(a+b) a^2-b^2=(a-b)(a+b) (a−b)2=a2−2ab+b2 (a-b)^2=a^2-2ab+b^2

(a+b)2=a2+2ab+b2 (a+b)^2=a^2+2ab+b^2

Remember that there might be more than one solution for the value of x.

According to the first formula:

x2=a2 x^2=a^2

We'll take the square root:

x=a x=a

25=b2 25=b^2

We'll take the square root:

b=5 b=5

We'll use the first shortened multiplication formula:

a2−b2=(a−b)(a+b) a^2-b^2=(a-b)(a+b)

x2−25=(x−5)(x+5)=0 x^2-25=(x-5)(x+5)=0

Therefore:

x+5=0 x+5=0

x=−5 x=-5

Or:

x−5=0 x-5=0

x=5 x=5

3

Final Answer

x=5,x=−5 x=5,x=-5

Key Points to Remember

Essential concepts to master this topic
  • Pattern: Recognize x2−25 x^2 - 25 as difference of squares formula
  • Factor: Apply a2−b2=(a−b)(a+b) a^2 - b^2 = (a-b)(a+b) to get (x−5)(x+5)=0 (x-5)(x+5) = 0
  • Check: Substitute both solutions: 52−25=0 5^2 - 25 = 0 and (−5)2−25=0 (-5)^2 - 25 = 0 ✓

Common Mistakes

Avoid these frequent errors
  • Finding only one solution instead of both
    Don't solve x2=25 x^2 = 25 by taking just the positive square root = missing x = -5! This ignores that both positive and negative numbers square to give positive results. Always remember that quadratic equations typically have two solutions.

Practice Quiz

Test your knowledge with interactive questions

Find the value of the parameter x.

\( 2x^2-7x+5=0 \)

FAQ

Everything you need to know about this question

Why are there two answers for this equation?

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Because both 5 and -5 when squared equal 25! When you have x2=25 x^2 = 25 , you need both the positive and negative square roots: x=±5 x = ±5 .

How do I know when to use the difference of squares formula?

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Look for the pattern a2−b2 a^2 - b^2 where both terms are perfect squares separated by a minus sign. Here, x2 x^2 and 25=52 25 = 5^2 fit this pattern perfectly!

Can I solve this by adding 25 to both sides instead?

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Yes! You can solve x2−25=0 x^2 - 25 = 0 by getting x2=25 x^2 = 25 , then taking the square root of both sides. Just remember the ± symbol for both solutions!

What if the equation was x² + 25 = 0?

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Then you'd get x2=−25 x^2 = -25 , which has no real solutions because you can't take the square root of a negative number in the real number system.

How do I check my answers?

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Substitute each solution back into the original equation. For x = 5: 52−25=25−25=0 5^2 - 25 = 25 - 25 = 0 ✓. For x = -5: (−5)2−25=25−25=0 (-5)^2 - 25 = 25 - 25 = 0 ✓

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