Question
Factor the following problem completely. \( 10 x^{2}-110 x+280 \)
Ask by Lang Valdez. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression \( 10x^{2} - 110x + 280 \) factors completely to \( 10(x - 4)(x - 7) \).
Solution
Step 1. Factor out the Greatest Common Factor (GCF).
The given expression is 10x² - 110x + 280. Notice that 10 is a common factor of all terms. Factor it out:
10(x² - 11x + 28)
Step 2. Factor the quadratic inside the parentheses.
We need to factor the quadratic x² - 11x + 28. Look for two numbers that multiply to +28 (the constant term) and add to -11 (the coefficient of x). The numbers -4 and -7 satisfy these conditions because:
(-4) × (-7) = 28
(-4) + (-7) = -11
Thus, the quadratic factors as:
(x - 4)(x - 7)
Step 3. Write the completely factored form.
Substitute the factors back into the expression with the GCF:
10(x - 4)(x - 7)
This is the completely factored form of the given expression.
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Simplify this solution Bonus Knowledge
To factor the expression \( 10x^{2} - 110x + 280 \) completely, start by taking out the greatest common factor, which is 10: \( 10(x^{2} - 11x + 28) \). Next, focus on factoring the quadratic \( x^{2} - 11x + 28 \). You need two numbers that multiply to 28 and add up to -11. These numbers are -4 and -7. Thus, you can write: \( x^{2} - 11x + 28 = (x - 4)(x - 7) \). Putting it all together, the complete factorization is: \( 10(x - 4)(x - 7) \). Voila! You've got the fully factored form.
