To determine which equations are true, we will simplify and verify each one step-by-step.
1. [tex]\( x^2 - 169 = (x - 13)(x + 13) \)[/tex]
- The left side of the equation is [tex]\( x^2 - 169 \)[/tex].
- The right side of the equation is [tex]\( (x - 13)(x + 13) \)[/tex], which simplifies to [tex]\( x^2 - 169 \)[/tex].
- Since both sides are equal, the equation [tex]\( x^2 - 169 = (x - 13)(x + 13) \)[/tex] is true.
2. [tex]\( 25x^2 - 12 = (5x - 6)(5x + 6) \)[/tex]
- The left side of the equation is [tex]\( 25x^2 - 12 \)[/tex].
- The right side of the equation is [tex]\( (5x - 6)(5x + 6) \)[/tex], which simplifies to [tex]\( 25x^2 - 36 \)[/tex].
- Since [tex]\( 25x^2 - 12 \neq 25x^2 - 36 \)[/tex], the equation [tex]\( 25x^2 - 12 = (5x - 6)(5x + 6) \)[/tex] is false.
3. [tex]\( 16x^2 - 18y^2 = (8x - 9y)(8x + 9y) \)[/tex]
- The left side of the equation is [tex]\( 16x^2 - 18y^2 \)[/tex].
- The right side of the equation is [tex]\( (8x - 9y)(8x + 9y) \)[/tex], which simplifies to [tex]\( 64x^2 - 81y^2 \)[/tex].
- Since [tex]\( 16x^2 - 18y^2 \neq 64x^2 - 81y^2 \)[/tex], the equation [tex]\( 16x^2 - 18y^2 = (8x - 9y)(8x + 9y) \)[/tex] is false.
4. [tex]\( 16x^2 - 64y^2 = (4x - 8y)(4x + 8y) \)[/tex]
- The left side of the equation is [tex]\( 16x^2 - 64y^2 \)[/tex].
- The right side of the equation is [tex]\( (4x - 8y)(4x + 8y) \)[/tex], which simplifies to [tex]\( 16x^2 - 64y^2 \)[/tex].
- Since both sides are equal, the equation [tex]\( 16x^2 - 64y^2 = (4x - 8y)(4x + 8y) \)[/tex] is true.
5. [tex]\( 9x^2 - 1 = (3x - 1)(3x + 1) \)[/tex]
- The left side of the equation is [tex]\( 9x^2 - 1 \)[/tex].
- The right side of the equation is [tex]\( (3x - 1)(3x + 1) \)[/tex], which simplifies to [tex]\( 9x^2 - 1 \)[/tex].
- Since both sides are equal, the equation [tex]\( 9x^2 - 1 = (3x - 1)(3x + 1) \)[/tex] is true.
Thus, the true equations are:
- [tex]\( x^2 - 169 = (x - 13)(x + 13) \)[/tex]
- [tex]\( 16x^2 - 64y^2 = (4x - 8y)(4x + 8y) \)[/tex]
- [tex]\( 9x^2 - 1 = (3x - 1)(3x + 1) \)[/tex]
