To factorise the quadratic expression [tex]\( 24 + 5x - x^2 \)[/tex], follow these detailed steps:
1. Identify the Standard Form: The expression is given in the form [tex]\( ax^2 + bx + c \)[/tex]. Here, it is [tex]\( -x^2 + 5x + 24 \)[/tex].
2. Reorder the Expression: For easier handling, write the quadratic term first:
[tex]\[ -x^2 + 5x + 24 \][/tex]
3. Factor Out the Leading Coefficient if Negative: Since the quadratic term has a negative coefficient, first factor out [tex]\(-1\)[/tex]:
[tex]\[ -1(x^2 - 5x - 24) \][/tex]
4. Factorise the Quadratic Inside the Parentheses: Now, we need to factorise the quadratic expression [tex]\( x^2 - 5x - 24 \)[/tex]. To do this, look for two numbers that multiply to the constant term [tex]\(-24\)[/tex] (c) and add up to the linear coefficient [tex]\(-5\)[/tex] (b).
The pair of numbers that work are [tex]\( -8 \)[/tex] and [tex]\( 3 \)[/tex] because:
[tex]\[ -8 \times 3 = -24 \][/tex]
[tex]\[ -8 + 3 = -5 \][/tex]
5. Write the Quadratic as a Product of Linear Factors: Using these numbers, factorise the quadratic:
[tex]\[ x^2 - 5x - 24 = (x - 8)(x + 3) \][/tex]
6. Include the Factored Out Negative: Substitute this back into the expression we had after factoring out [tex]\(-1\)[/tex]:
[tex]\[ -1(x^2 - 5x - 24) \implies -1(x - 8)(x + 3) \][/tex]
7. Final Factored Form: Therefore, the complete factorisation of the original expression [tex]\( 24 + 5x - x^2 \)[/tex] is:
[tex]\[ -(x - 8)(x + 3) \][/tex]
So, the factorised form of [tex]\( 24 + 5x - x^2 \)[/tex] is [tex]\( -(x - 8)(x + 3) \)[/tex].
