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math questions help!!!!

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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
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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
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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