Question 143273
Do you want to factor?








Looking at {{{x^2-16x+63}}} we can see that the first term is {{{x^2}}} and the last term is {{{63}}} where the coefficients are 1 and 63 respectively.


Now multiply the first coefficient 1 and the last coefficient 63 to get 63. Now what two numbers multiply to 63 and add to the  middle coefficient -16? Let's list all of the factors of 63:




Factors of 63:

1,3,7,9,21,63


-1,-3,-7,-9,-21,-63 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 63

1*63

3*21

7*9

(-1)*(-63)

(-3)*(-21)

(-7)*(-9)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -16


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">63</td><td>1+63=64</td></tr><tr><td align="center">3</td><td align="center">21</td><td>3+21=24</td></tr><tr><td align="center">7</td><td align="center">9</td><td>7+9=16</td></tr><tr><td align="center">-1</td><td align="center">-63</td><td>-1+(-63)=-64</td></tr><tr><td align="center">-3</td><td align="center">-21</td><td>-3+(-21)=-24</td></tr><tr><td align="center">-7</td><td align="center">-9</td><td>-7+(-9)=-16</td></tr></table>



From this list we can see that -7 and -9 add up to -16 and multiply to 63



Now looking at the expression {{{x^2-16x+63}}}, replace {{{-16x}}} with {{{-7x+-9x}}} (notice {{{-7x+-9x}}} adds up to {{{-16x}}}. So it is equivalent to {{{-16x}}})


{{{x^2+highlight(-7x+-9x)+63}}}



Now let's factor {{{x^2-7x-9x+63}}} by grouping:



{{{(x^2-7x)+(-9x+63)}}} Group like terms



{{{x(x-7)-9(x-7)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-9}}} out of the second group



{{{(x-9)(x-7)}}} Since we have a common term of {{{x-7}}}, we can combine like terms


So {{{x^2-7x-9x+63}}} factors to {{{(x-9)(x-7)}}}



So this also means that {{{x^2-16x+63}}} factors to {{{(x-9)(x-7)}}} (since {{{x^2-16x+63}}} is equivalent to {{{x^2-7x-9x+63}}})




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     Answer:

So {{{x^2-16x+63}}} factors to {{{(x-9)(x-7)}}}