Answer:
i) [tex](x-3)(x + 8)[/tex]
ii) [tex]x = -8[/tex] and [tex]x = 3[/tex]
Step-by-step explanation:
i) To factorize [tex]x^2 + 5x - 24[/tex], we need to find two numbers that multiply to give [tex]24[/tex] and add to give [tex]5[/tex].
These numbers are [tex]8[/tex] and [tex]-3[/tex] because [tex]8 \times (-3) = -24[/tex] and [tex]8 + (-3) = 5[/tex].
So, we can rewrite the expression as:
[tex]x^2 + (8-3)x - 24[/tex]
[tex]x^2 + 8x - 3x - 24[/tex]
Now, we group the terms and factor by grouping:
[tex]x(x + 8) - 3(x + 8)[/tex]
Now, we can factor out the common factor [tex](x + 8)[/tex]:
[tex](x + 8)(x - 3)[/tex]
So, [tex]x^2 + 5x - 24[/tex] factors to [tex](x -3)(x +8)[/tex].
ii) To solve [tex]x^2 + 5x - 24 = 0[/tex], we can use the factored form from part (i) and set each factor equal to zero:
[tex](x - 3)(x + 8) = 0[/tex]
Setting each factor equal to zero:
[tex]x - 3 = 0[/tex] or [tex]x + 8 = 0[/tex]
Solving each equation:
For [tex]x + 8 = 0[/tex]:
[tex]x = -8[/tex]
For [tex]x - 3 = 0[/tex]:
[tex]x = 3[/tex]
So, the solutions to [tex]x^2 + 5x - 24 = 0[/tex] are [tex]x = -8[/tex] and [tex]x = 3[/tex].
