# math problem?

For how many positive integers k less than 2020 does the equation below have real solutions for x?

(2x+1)(x+k)=(3x–2)(x−k)

### 4 Answers

- Ian HLv 71 month ago
(3x – 2)(x − k) - (2x + 1)(x + k) = x^2 – (5k + 3)x + k = 0

Real x if y = (5k + 3)^2 - 4k = 25k^2 + 26k + 9 ≥ 0

https://www.wolframalpha.com/input/?i=y+%3D+25k%5E...

That graph shows that y = 25k^2 + 26k + 9 ≥ 0 is positive for all k

So, all positive integers k less than 2020 have real solutions for x

Examples:

k = 100, x ~ 502.801 or x ~ 0.198886

k = 1000, x ~ 5002.8 or x ~ 0.199888

k = 2000, x ~ 10002.8 or x ~ 0.199944

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- PopeLv 71 month ago
It is a quadratic equation. Expand it and group like terms.

(2x + 1)(x + k) = (3x - 2)(x - k)

2x² + (2k + 1)x + k = 3x² + (-3k - 2)x + 2k

x² + (-5k - 3)x + k = 0

The equation has real solutions if and only if the discriminant is greater than or equal to zero.

discriminant ≥ 0

(-5k - 3)² - 4(1)(k) ≥ 0

25k² + 30k + 9 - 4k ≥ 0

25k² + 26k + 9 ≥ 0

A second discriminant, that of 25k² + 26k + 9, is -224.

Therefore, 25k² + 26k + 9 has no real roots.

The leading coefficient of 25k² + 26k + 9 is positive.

Therefore, 25k² + 26k + 9 > 0 for all real k.

Thus the given equation must have real solutions for all real k.

There are 2019 positive integers less than 220. That is your number, 2019.

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- davidLv 71 month ago
(2x+1)(x+k) = (3x–2)(x−k)

2x^2 + x + 2kx + k = 3x^2 - 2x - 3kx + 2k

x^2 - 5kx - 3x + k = 0

b = -(5k + 3)

a = 1 .. c = k

discriminant

b^2 - 4ac = (5k + 3)^2 - 4(1)(k)

= 25k^2 + 30k + 9 - 4k

= 25k^2 + 26k + 9

... when k is positive, the discriminant is always positive which means ther will always be real solutions

... all k < 2020 will give real solutions for x

. so 2019 posivie valies of K create real solutions for x

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- PhilipLv 61 month ago
k is an integer. 0 < k < 2020.

(3x-2)(x-k)-(2x+1)(x+k) = 0, ie.,

x^2[3-2]+x[-2-3k-2k-1]+k[2-1] =0, ie., x^2-8kx+k=0

Then 2x=8k(+/-)D, where D^2 = (8k)^2-4k. For

real roots we require D^2(=/>)0, ie., require

4k(16k-1)(=/>)0, which holds for all k in [1,2019].

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