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Question 1: What is the product of all constants \(k \) such that the quadratic \(x^2 + kx +15\) can be factored in the form \((x+a)(x+b)\), where a and b are integers?

Question 2: Find all values of x such that \(\dfrac{x}{x+4} = -\dfrac{9}{x+3}\) .

Question 3: Find all real numbers \(x\) that satisfy the equation \((x-4)(x-8) = 12\).

Question 4: Find all real values of  \(x\) that satisfy the equation: \( x^2 - 7x = 98. \)

 Dec 2, 2018
 #1
avatar+17332 
+2

The numbers   a   and  b   must multiply to 15    and add to k

a   b        can be        added =k

1  15                             16

3   5                                8

-1  -15                           -16

-3   -5                              -8

 

all of the k's multiplied results in   16384

 

2

Cross multiply to get

x^2 +3x = -9x-36

x^2+12x+36=0

(x+6)(x+6) = 0    x = - 6

      

 

3

x^2 -12x +32 -12 = 0

    x^2-12x+20 = 0

    (x-10)(x-2) = 0     so x = 10 or 2

 

4

x^2-7x -98 =0    use quadratic formula to find x = 14  or -7

 Dec 2, 2018
edited by Guest  Dec 2, 2018
edited by Guest  Dec 2, 2018
 #2
avatar+3994 
+2

3. Let's try by completing the square. We simplify the original expression, to \(x^2-12x=-20\). Then, solve for \(a\) while completing the square, so \(2ax=-12x, a=-6\) . Finally, you end up with \(\left(x-6\right)^2=16\), therefore the only solutions are \(\boxed{x=2, x=10}\) .

 Dec 2, 2018

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