Factorize the following algebraic in \( 7 x^{2}-63 \)

To factor the expression \( 7x^{2} - 63 \), start by identifying the greatest common factor (GCF), which is 7. Factoring out the GCF gives us: \[ 7(x^{2} - 9) \] Now, notice that \( x^{2} - 9 \) is a difference of squares, which can be further factored as: \[ 7(x - 3)(x + 3) \] So, the fully factored form of \( 7x^{2} - 63 \) is \( 7(x - 3)(x + 3) \). Factoring isn't just an academic exercise; it has real-world applications in areas like physics and engineering, where equations are often simplified to make calculations easier. By breaking down complex equations into manageable parts, professionals can solve problems more efficiently and effectively, ensuring better outcomes in design and analysis! Misplacing a negative sign or skipping steps in the factorization process can lead to incorrect results. Always double-check your work, especially with the difference of squares, which can be tricky! A fun tip is to remember the pattern: \( a^2 - b^2 = (a - b)(a + b) \) – visualize it as two kids tugging at opposite ends of a rope, each ready to pull in their own direction!