Which expression represents -8×2 + 54x + 140 in factored form?
The correct answer and explanation is:
To factor the expression −8×2+54x+140-8x^2 + 54x + 140, let’s go through the process step by step.
Step 1: Factor out the greatest common factor (GCF)
The first thing to do is to identify the GCF of all the terms in the expression. The terms are −8×2-8x^2, 54x54x, and 140140.
- The GCF of −8-8, 5454, and 140140 is 2 (we can factor out the negative sign later if needed).
- Let’s factor out 2 from the entire expression:
−8×2+54x+140=−2(4×2−27x−70)-8x^2 + 54x + 140 = -2(4x^2 – 27x – 70)
Now, we need to focus on factoring the quadratic expression 4×2−27x−704x^2 – 27x – 70.
Step 2: Factor the quadratic expression
Next, we factor 4×2−27x−704x^2 – 27x – 70. To factor this, we will look for two numbers that multiply to 4×−70=−2804 \times -70 = -280 and add up to −27-27.
The pair of numbers that works are 77 and −40-40, since: 7×(−40)=−280and7+(−40)=−277 \times (-40) = -280 \quad \text{and} \quad 7 + (-40) = -27
Step 3: Split the middle term
Now, we can split the middle term −27x-27x using 7x7x and −40x-40x: 4×2−27x−70=4×2+7x−40x−704x^2 – 27x – 70 = 4x^2 + 7x – 40x – 70
Step 4: Factor by grouping
Now, group the terms in pairs: (4×2+7x)−(40x+70)(4x^2 + 7x) – (40x + 70)
Factor each group: x(4x+7)−10(4x+7)x(4x + 7) – 10(4x + 7)
Step 5: Factor out the common binomial
Now, factor out the common binomial factor 4x+74x + 7: x(4x+7)−10(4x+7)=(x−10)(4x+7)x(4x + 7) – 10(4x + 7) = (x – 10)(4x + 7)
Step 6: Combine with the factor outside the parentheses
Now, don’t forget the −2-2 that we factored out in Step 1. So the fully factored form of the expression is: −2(x−10)(4x+7)-2(x – 10)(4x + 7)
Final Answer:
The expression −8×2+54x+140-8x^2 + 54x + 140 in factored form is: −2(x−10)(4x+7)-2(x – 10)(4x + 7)
Explanation:
To summarize the process, we first identified the greatest common factor (GCF) and factored that out, reducing the quadratic expression to a more manageable form. Then we used factoring by grouping to split and combine the terms. The result was a factored form that reveals the roots of the equation, which are the values of xx that make the expression equal to zero when set equal to zero.