In the equation x^2 + 10x + 24 = (x + a)(x + b), b is an integer. Find algebraically all possible

1 answer:

8 0

To factor (x² +10x+24); you first have to ask yourself, what 2 numbers will multiply and give me positive 24, but add and give me positive 10? After listing your factors, (1,24)(2,12)(3,8)and (4,6); you decide that (4*6)=24 & adds to get you 10. Meaning that (x²+10x+24) factors are (x+4)(x+6).
Now algebraically, there wouldn't be any other possible value of b, 6, because if you put (-6) & (4), those numbers will multiply and give you (-24) and add & give you (-2); & that is not what you're looking for.

I hope I explained this enough for you to understand. (:

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If log6⁡⁡3+log6⁡⁡72=x, what is the value of x?

soldi70 [24.7K]

$\bf \textit{logarithm of factors}\\\\
log_{{ a}}(xy)\implies log_{{ a}}(x)+log_{{ a}}(y)
\\\\\\
\textit{Logarithm Change of Base Rule}\\\\
log_{{ a}}{{ b}}\implies \cfrac{log_{{ c}}{{ b}}}{log_{{ c}}{{ a}}}\\\\
-------------------------------\\\\
log_6(3)+log_6(72)=x\implies log_6(3\cdot 72)=x\implies log_6(216)=x
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recall that, log <--- with no apparent base, implies base10, so you can just plug that in your calculator

for the change of base rule, it doesn't really matter what base you use, so long is the same above and below, it just so happen, that we used base10 in this case, but could have been anything, same result.

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