Fill the Blanks: Solve the Quadratic Equation (_+3)·(_-3) = x²-9

Difference of Squares with Blank Expressions

Fill in the missing element to obtain a true expression:

(—+3)⋅(—−3)=x2−9 (_—+3)\cdot(_—-3)=x^2-9

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Complete the missing value
00:03 We'll use the shortened multiplication formulas
00:07 Let's substitute X as A
00:11 and 9 as B
00:16 We'll extract the root and find A
00:22 A is the missing element
00:26 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

Fill in the missing element to obtain a true expression:

(—+3)⋅(—−3)=x2−9 (_—+3)\cdot(_—-3)=x^2-9

2

Step-by-step solution

To solve this problem, let's use the difference of squares formula, which is (a+b)(a−b)=a2−b2 (a + b)(a - b) = a^2 - b^2 . Given the equation (+3)(−3)=x2−9(_ + 3)(_- 3) = x^2 - 9, we can compare it to the formula:

  • a2=x2 a^2 = x^2 implies a=x a = x .
  • b2=9 b^2 = 9 implies b=3 b = 3 .

This means the expression (+3)(−3)(_ + 3)(_- 3) should represent (x+3)(x−3)(x + 3)(x - 3), satisfying the equation through the difference of squares formula.

Thus, the missing element to obtain a correct expression is x x .

3

Final Answer

x x

Key Points to Remember

Essential concepts to master this topic
  • Formula: (a+b)(a−b)=a2−b2 (a + b)(a - b) = a^2 - b^2 identifies missing terms
  • Technique: Compare x2−9 x^2 - 9 to a2−b2 a^2 - b^2 where a=x a = x , b=3 b = 3
  • Check: Expand (x+3)(x−3)=x2−9 (x + 3)(x - 3) = x^2 - 9 using FOIL method ✓

Common Mistakes

Avoid these frequent errors
  • Guessing without using the difference of squares formula
    Don't just guess what fits in the blanks without mathematical reasoning = wrong patterns! This leads to answers like y or √x that don't create the correct algebraic identity. Always identify the formula first, then match coefficients systematically.

Practice Quiz

Test your knowledge with interactive questions

Solve:

\( (2+x)(2-x)=0 \)

FAQ

Everything you need to know about this question

How do I know this is a difference of squares problem?

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Look for the pattern a2−b2 a^2 - b^2 on the right side! Here, x2−9 x^2 - 9 matches this pattern since 9 = 3². This tells you to use the difference of squares formula.

Why can't the answer be y or √x?

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Let's check: (y+3)(y−3)=y2−9 (y + 3)(y - 3) = y^2 - 9 , not x2−9 x^2 - 9 ! And (x+3)(x−3)=x−9 (\sqrt{x} + 3)(\sqrt{x} - 3) = x - 9 , also wrong. Only x gives the correct result.

What if I don't remember the difference of squares formula?

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You can always expand using FOIL method: (x+3)(x−3)=x2−3x+3x−9=x2−9 (x + 3)(x - 3) = x^2 - 3x + 3x - 9 = x^2 - 9 . The middle terms cancel out, which is why it's called difference of squares!

How do I identify what goes in each blank?

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Both blanks must be the same variable or expression. Since the right side has x2 x^2 , both blanks need x to produce that term when multiplied together.

Can I work backwards from the answer?

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Absolutely! Start with x2−9 x^2 - 9 and think: "What two factors multiply to give this?" Since 9=32 9 = 3^2 , you get (x+3)(x−3) (x + 3)(x - 3) .

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