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avatar+144 

1. Find the constant \(p\) such that \(x^2 - 5x - 2x + p\) is the square of a binomial.

 

2. Find the constant \(k\) such that the quadratic \(2x^2 + 3x + 8x + k\) has a double root.

 

3. The roots of \(Ax^2+Bx+1\) are the same as the roots of \(x^2- 3x + 5\). What is \(A+B\)?

 

4. Find the largest integer \(k\) such that the equation \(5x^2 - kx + 8 - 3x^2 + 15 = 0\) has no real solutions.

 

Any help is appreciated... thanks a lot.

 Feb 16, 2024
 #2
avatar+128772 
+2

1.   x^2 - 7x + p

 

p =  (7/2)^2  =  49/4

 

Factroization   is ( x - 7/2)^2

 

 

cool cool cool

 Feb 16, 2024
 #3
avatar+128772 
+2

2.   2x^2 + 11x + k

 

We will have a double root when  the discriminant =   0

 

So

 

11^2 - 4 * 2 * k    = 0

 

121 - 8k   = 0

 

121  = 8k

 

k = 121/8

 

cool cool cool

 Feb 16, 2024
 #4
avatar+144 
+2

That was so fast... they're both correct, thanks so much!!!

Now the harder ones indecision

 Feb 16, 2024
 #5
avatar+128772 
+2

3.   x^2 - 3x  + 5  = 0 

 

Note that if we  just  divide through by 5   we get

 

(1/5)x^2  - (3/5)x + 1  = 0

 

A =1/5     B  =(-3/5)

 

A + B  =  1/5    -3/5 =  -2/5

 

{You can check to see that the roots are the same }

 

cool cool cool

 Feb 16, 2024
 #6
avatar+128772 
+2

4.  Simplified we get

 

2x^2 -kx + 23  = 0

 

We have no real solutions when the discriminant is <  0

 

So

 

k^2  - 4(2)(23) < 0

 

k^2  -  184  < 0

 

k^2  <  184

 

k <  sqrt (184)   ≈ 13.564

 

So  the largest integer =  13

 

 

cool cool cool

 Feb 16, 2024
 #7
avatar+144 
+2

Thank you so much CPhill!!!!

(that took you like no time at all surprise)

 Feb 16, 2024
 #8
avatar+128772 
+1

You're Welcome  !!!

 

 

cool cool cool

CPhill  Feb 16, 2024

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