How do you factor the following equation?

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How do you factor the following equation?

$2 x^{2} + 10 x + 84$ $2x^2 + 10x + 84$ .

1 Answer

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Answer 1 · 2017-09-05T00:44:55

If we are restricted to Real factors: $2 (x^{2} + 5 x + 42)$ $color(red)(2(x^2+5x+42))$
If Complex factors are allowed: $2 (x + \frac{5 - \sqrt{143}}{2}) (x + \frac{5 + \sqrt{143}}{2})$ $color(blue)(2(x+(5-sqrt(143))/2)(x+(5+sqrt(143))/2))$

Explanation:

The terms of $2 x^{2} + 10 x + 84$ $2x^2+10x+84$
have the obvious constant common factor $2$ $2$
and therefore can b factored as $2 (x^{2} + 5 x + 42)$ $2(x^2+5x+42)$

The determinant of $(x^{2} + 5 x + 42)$ $(x^2+5x+42)$ is negative so there are no Real factors for this expression.
$XXXX$ $color(white)("XXXX")$ Side note
$XXXX$ $color(white)("XXXX")$ for an expression of the form $a x^{2} + b x + c$ $ax^2+bx+c$
$XXXX$ $color(white)("XXXX")$ the determinant is $Delta=b^2-4ac$
$color(white)("XXXX")$ and it is a general rule that $Delta < 0$
$color(white)("XXXX")$ implies no possible (Real) factors

However if Complex values are allowed then a factoring can be achieved using the quadratic formula.
$color(white)("XXXX")$ Side note 2
$color(white)("XXXX")$ The quadratic formula says that
$color(white)("XXXX")$ an expression of the form $ax^2+bx+c$
$color(white)("XXXX")$ can be factored with a term $(x-phi)$
$color(white)("XXXX")$ where $phi = (-b+-sqrt(b^2-4ac))/(2a)$
Plugging in the obvious values for $a, b, c$ from $1x^2+5x+42$ gives the second answer as shown above.

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How do you factor the following equation?

$2 x^{2} + 10 x + 84$ $2x^2 + 10x + 84$ .

1 Answer

Explanation:

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How do you factor the following equation?

2x^2 + 10x + 842x2+10x+842x^2 + 10x + 84.

1 Answer

Explanation:

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$2 x^{2} + 10 x + 84$ $2x^2 + 10x + 84$ .