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The expression factors to \( -4(2x - 5)(3x + 5) \).
Solution
We want to factor the quadratic expression
–24x² + 20x + 100.
Step 1. Factor out the greatest common factor (GCF). Notice that each coefficient is divisible by 4. Factoring out –4 (which makes the x² term positive inside the parentheses) gives:
–24x² + 20x + 100 = –4(6x² – 5x – 25).
Step 2. Factor the quadratic 6x² – 5x – 25. Multiply the coefficient of x² (6) by the constant term (–25) to get –150. We need two numbers that multiply to –150 and add to –5. The numbers 10 and –15 work because 10 × (–15) = –150 and 10 + (–15) = –5.
Step 3. Rewrite the middle term –5x using 10x and –15x:
6x² – 5x – 25 = 6x² + 10x – 15x – 25.
Step 4. Factor by grouping:
Group the terms:
(6x² + 10x) + (–15x – 25).
Factor out the common factors in each group:
2x(3x + 5) – 5(3x + 5).
Now, factor out the common binomial (3x + 5):
(3x + 5)(2x – 5).
Step 5. Substitute back into the expression with the –4 factored out:
–24x² + 20x + 100 = –4(2x – 5)(3x + 5).
Thus, the completely factored form is:
–4(2x – 5)(3x + 5).
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Simplify this solution Bonus Knowledge
Did you know that quadratic equations like \( -24x^{2} + 20x + 100 \) can represent the path of an object in motion? For instance, if this equation described the height of an object over time, the vertex of the parabola could tell you the maximum height the object reaches! When solving such equations, remember to factor or use the quadratic formula cautiously. A common mistake is to overlook the negative leading coefficient, which indicates the parabola opens downwards. This can lead to confusion about the nature of roots and the direction of the graph. Always double-check your signs!
