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# Algebra

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If $s$ is a real number, then what is the smallest possible value of $2s^2 - 8s + 19 - 7s^2 - 8s + 20$?

Apr 9, 2024

#1
+9665
+1

I assume you meant largest.

Simplifying the expression and completing the square, we have

$$2s^2 - 8s + 19 - 7s^2 - 8s + 20\\ = -5s^2 - 16s + 39\\ = -5(s^2 + \frac{16}5s) + 39\\ = -5(s + \frac85)^2 + 5(\frac 85)^2 + 39\\ =-5(s + \frac85)^2 + \frac{259}5$$

Note that $$-5(s + \frac85)^2 \leq 0$$ for any real s. Then the largest possible value is $$\dfrac{259}5$$.

There is no smallest value. The expression can be arbitrarily small.

Apr 9, 2024

#1
+9665
+1

I assume you meant largest.

Simplifying the expression and completing the square, we have

$$2s^2 - 8s + 19 - 7s^2 - 8s + 20\\ = -5s^2 - 16s + 39\\ = -5(s^2 + \frac{16}5s) + 39\\ = -5(s + \frac85)^2 + 5(\frac 85)^2 + 39\\ =-5(s + \frac85)^2 + \frac{259}5$$

Note that $$-5(s + \frac85)^2 \leq 0$$ for any real s. Then the largest possible value is $$\dfrac{259}5$$.

There is no smallest value. The expression can be arbitrarily small.

MaxWong Apr 9, 2024