document.write( "Question 1026782: If s is a real number, then what is the smallest possible value of 2s^2 - 8s + 19? \n" ); document.write( "

Algebra.Com's Answer #642089 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2s%5E2-8s%2B19\"$ is a quadratic polynomial.
\n" ); document.write( "Quadratic functions such as $\"f%28x%29=ax%5E2%2Bbx%2Bc\"$ with {{a<>0}}} ,
\n" ); document.write( "have a maximum (if $\"a%3C0\"$ ),
\n" ); document.write( "or a minimum (if $\"a%3E0\"$ ).
\n" ); document.write( "So, since the coefficient of $\"s%5E2\"$ is $\"2%3E0\"$ ,
\n" ); document.write( " $\"2s%5E2-8s%2B19\"$ has a minimum value}}} .
\n" ); document.write( "
\n" ); document.write( "How could you find that minimum value?
\n" ); document.write( "You could remember and apply a recipe handed down by your algebra 2 teacher.
\n" ); document.write( "Alternatively, you could use your knowledge of algebra, and your ability to think.
\n" ); document.write( "
\n" ); document.write( "APPLYING MEMORIZED FORMULAS/RECIPES:
\n" ); document.write( "A quadratic function {{f(x)=ax^2+bx+c}}} with $\"a%3C0\"$ has a minimum for $\"X=%28-b%29%2F%222+a%22\"$ .
\n" ); document.write( "In the case of $\"2s%5E2-8s%2B19\"$ ,
\n" ); document.write( "the variable is $\"s\"$ rather than $\"x\"$ , $\"a=2\"$ and $\"b=-8\"$ ,
\n" ); document.write( "so the minimum happens for $\"s=%28-%28-8%29%29%2F%282%2A2%29=8%2F4=2\"$ ,
\n" ); document.write( "and for $\"s=2\"$ , $\"2s%5E2-8s%2B19=2%2A2%5E2-8%2A2%2B19=2%2A4-16%2B19=8-16%2B19=highlight%2811%29\"$ .
\n" ); document.write( "
\n" ); document.write( "IF YOU DID NOT MEMORIZE THAT FORMULA,
\n" ); document.write( "you have to reason your way to the answer.
\n" ); document.write( " $\"2s%5E2-8s=2%28s%5E2-4s%29\"$ may remind you of $\"s%5E2-4s%2B4=%28s-2%29%5E2\"$ ,
\n" ); document.write( "and $\"2%28s-2%29%5E2=2%28s%5E2-4s%2B4%29=2s%5E2-8s%2B8\"$ .
\n" ); document.write( "So, you can complete the square:
\n" ); document.write( " $\"2s%5E2-8s%2B19=2s%5E2-8s%2B8%2B19-8=2%28s-2%29%5E2%2B19-8=2%28s-2%29%5E2%2B11\"$ .
\n" ); document.write( "You know that the square of any real number is non-negative,
\n" ); document.write( "so $\"%28s-2%29%5E2%3E=0\"$ , and $\"2s%5E2-8s%2B19=2%28s-2%29%5E2%2B11%3E=0%2B11=11\"$ ,
\n" ); document.write( "which means the the smallest possible value of $\"2s%5E2-8s%2B19\"$ is $\"highlight%2811%29\"$ . \n" ); document.write( "

\n" );