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: \( x^2 - 7x = 98. \)
+36916 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
+4609 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 \(\boxed{x=2, x=10}\) .
