I did this even easier and it may not have been ironclad but it worked– I just wasked myself what value of a will give me the expression on the leftside of the equation. I figure 3 x 8 would give me 24. Then I just multiplied it out and when I saw the 17 show up next to the x cooresponding with be I knew this could be the answer.
(3x+8)( x+3) = 3x^2 +17x +24, so I figure that 17 was the answer with a as three it satisfies the equation only if 17 is the coefficient in front of x. This took me about 2 minutes to solve. It is not solving the problem that counts its how long you can do it in. Any problem that takes longer than 2 minutes means I dont have the concept mastered or I have errored and must learn to move on.
I just did it this way; personally I think it’s easier but different people have different methods!
Since the question tells us “…is true for all values of x”, then x can be any number. I chose the number zero because anything multiplied by zero gets you zero! If you substitute zero for all x’s in the equation, you’re left with 24 = (8)(a) or just 24 = 8a. So, a = 3.
Now, substitute 3 (since a = 3) for a on the right side of the equation: 3x^2 + bx + 24 = (3x+8)(x+3). Now all you have to do it foil to find b!
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I did this even easier and it may not have been ironclad but it worked– I just wasked myself what value of a will give me the expression on the leftside of the equation. I figure 3 x 8 would give me 24. Then I just multiplied it out and when I saw the 17 show up next to the x cooresponding with be I knew this could be the answer.
(3x+8)( x+3) = 3x^2 +17x +24, so I figure that 17 was the answer with a as three it satisfies the equation only if 17 is the coefficient in front of x. This took me about 2 minutes to solve. It is not solving the problem that counts its how long you can do it in. Any problem that takes longer than 2 minutes means I dont have the concept mastered or I have errored and must learn to move on.
I just did it this way; personally I think it’s easier but different people have different methods!
Since the question tells us “…is true for all values of x”, then x can be any number. I chose the number zero because anything multiplied by zero gets you zero! If you substitute zero for all x’s in the equation, you’re left with 24 = (8)(a) or just 24 = 8a. So, a = 3.
Now, substitute 3 (since a = 3) for a on the right side of the equation: 3x^2 + bx + 24 = (3x+8)(x+3). Now all you have to do it foil to find b!
Final answer: b = 17