```Question 376723

Looking at the expression {{{2h^2-19h+24}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-19}}}, and the last term is {{{24}}}.

Now multiply the first coefficient {{{2}}} by the last term {{{24}}} to get {{{(2)(24)=48}}}.

Now the question is: what two whole numbers multiply to {{{48}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-19}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{48}}} (the previous product).

Factors of {{{48}}}:

1,2,3,4,6,8,12,16,24,48

-1,-2,-3,-4,-6,-8,-12,-16,-24,-48

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to {{{48}}}.

1*48 = 48
2*24 = 48
3*16 = 48
4*12 = 48
6*8 = 48
(-1)*(-48) = 48
(-2)*(-24) = 48
(-3)*(-16) = 48
(-4)*(-12) = 48
(-6)*(-8) = 48

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-19}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>48</font></td><td  align="center"><font color=black>1+48=49</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>2+24=26</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>16</font></td><td  align="center"><font color=black>3+16=19</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>4+12=16</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>6+8=14</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-48</font></td><td  align="center"><font color=black>-1+(-48)=-49</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>-2+(-24)=-26</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>-16</font></td><td  align="center"><font color=red>-3+(-16)=-19</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>-4+(-12)=-16</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>-6+(-8)=-14</font></td></tr></table>

From the table, we can see that the two numbers {{{-3}}} and {{{-16}}} add to {{{-19}}} (the middle coefficient).

So the two numbers {{{-3}}} and {{{-16}}} both multiply to {{{48}}} <font size=4><b>and</b></font> add to {{{-19}}}

Now replace the middle term {{{-19h}}} with {{{-3h-16h}}}. Remember, {{{-3}}} and {{{-16}}} add to {{{-19}}}. So this shows us that {{{-3h-16h=-19h}}}.

{{{2h^2+highlight(-3h-16h)+24}}} Replace the second term {{{-19h}}} with {{{-3h-16h}}}.

{{{(2h^2-3h)+(-16h+24)}}} Group the terms into two pairs.

{{{h(2h-3)+(-16h+24)}}} Factor out the GCF {{{h}}} from the first group.

{{{h(2h-3)-8(2h-3)}}} Factor out {{{8}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

{{{(h-8)(2h-3)}}} Combine like terms. Or factor out the common term {{{2h-3}}}

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So {{{2h^2-19h+24}}} factors to {{{(h-8)(2h-3)}}}.

In other words, {{{2h^2-19h+24=(h-8)(2h-3)}}}.

Note: you can check the answer by expanding {{{(h-8)(2h-3)}}} to get {{{2h^2-19h+24}}} or by graphing the original expression and the answer (the two graphs should be identical).

If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help">jim_thompson5910@hotmail.com</a>

Also, feel free to check out my <a href="http://www.freewebs.com/jimthompson5910/home.html">tutoring website</a>

Jim```