I need help factoring this x^2 + 5x + 8

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degeneratecircle's profile pic

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

This just goes to show that "factoring `x^2+5x+8`" is ambiguous, and we need to know which factors are allowed. For example,

`x^2-2`

can't be factored over the integers because we can't write

`x^2-2=(ax-b)(cx-d)` where `a,b,c,d` are integers. However, we can factor the same expression over the real numbers as

`x^2-2=(x-sqrt(2))(x+sqrt(2)).`

Likewise, `x^2+5x+8` can't be factored over the real numbers, but it can be factored over the complex numbers. You're probably the best judge on what's meant by the question (if you've covered complex numbers, then you probably want aruv's factorization, otherwise the first two answers are probably what you want).

baxthum8's profile pic

baxthum8 | High School Teacher | (Level 3) Associate Educator

Posted on

Factor `x^2 + 5x + 8`

In order to factor this expression, we need to find 2 factors of 8 that have a sum of 5.

`(x+a)(x+b) = x^2 + ax+bx + ab`

There are no factors of 8 that will make a sum of 5

`2*4=8 , 2+4=6`

`1*8=8, 1+8 = 9`

This expression can not be factored.

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The expression x^2 + 5x + 8 has to be factored. It is not possible to express 5 as a sum of two numbers a and b such that a*b = 8.

As a result, the expression x^2 + 5x + 8 cannot be factored.

atyourservice's profile pic

atyourservice | Student, Grade 11 | (Level 3) Valedictorian

Posted on

`x^2 + 5x + 8`

`a=1`    `b=5`     `c=8`

First you multiply a by c 

`1xx8=8`

now find factors of 8 that add to 5, there is no such thing so this problem can't be factored by real numbers

aruv's profile pic

aruv | High School Teacher | (Level 2) Valedictorian

Posted on

`x^2+5x+8=x^2+2(5/2)x+(5/2)^2-(5/2)^2+8`

`=(x+5/2)^2+7/4`

`=(x+5/2)^2-(i^2 7)/4`

`=(x+5/2)^2-((isqrt(7))/2)^2`

`=(x+(5+isqrt(7))/2)(x+(5-isqrt(7))/2)`

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