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Brandy is trying to factor the quadratic $3x^2 - x - 10.$ She starts by assuming that the quadratic factors as \[3x^2 - x - 10 = (x + B)(3x + D),\]for some integers $B$ and $D.$ After some work, Brandy successfully factors the quadratic. Find the ordered pair $(B,D).$

Respuesta :

Answer:

(B , D) = (-2,5)

Step-by-step explanation:

The given quadratic expression is

[tex]3x^{2} -x-10[/tex]

compare the given expression with [tex]ax^{2} +bx+c[/tex]

so we have [tex]a = 3 , b = -1 ,c = -10[/tex]

now [tex]ac=3 (-10)=- 30[/tex] and [tex]b= -1[/tex]

we need to find two numbers such that their product is -30 and sum is -1

-6 and 5 are such numbers

so we have

[tex]3x^{2} -6x+5x-10[/tex]

grouping first two terms and last two terms

[tex](3x^{2} -6x)+(5x-10)[/tex]

factor out 3x from first two terms and 5 from last two terms

[tex]3x(x-2) +5(x-2)[/tex]          

[tex](x-2)(3x+5)[/tex]        ( factor out (x-2))

[tex](x+(-2))(3x+5)[/tex]

hence B =-2 and D= 5

(B,D)= (-2,5)

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