I choose a random integer $n$ $n$ between $1$ $1$ and $10$ $10$ inclusive. What is the probability that for the $n$ $n$ I chose, there exist no real solutions to the equation $x (x + 5) = - n$ $x(x+5) = -n$ ? Express your answer as a common fraction.

Question

I choose a random integer $n$ $n$ between $1$ $1$ and $10$ $10$ inclusive. What is the probability that for the $n$ $n$ I chose, there exist no real solutions to the equation $x (x + 5) = - n$ $x(x+5) = -n$ ? Express your answer as a common fraction.

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Subject: Algebra
Focus: Percents
*Review Problem

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Answer 1 · 2018-06-14T07:39:14

$\frac{2}{5}$ $color(blue)(2/5)$

Explanation:

$x (x + 5) = - n$ $x(x+5)=-n$

Expand:

$x^{2} + 5 x + n = 0$ $x^2+5x+n=0$

We need the solutions to this equation to be non-real.

Using the discriminant of a quadratic:

$bb(b^2-4ac)$

If:

$b^2-4ac> 0$ The equation has two real solutions.

$b^2-4ac= 0$ The equation has repeated real solutions.

$b^2-4ac< 0$ The equation has non-real solutions.

We need the last situation.

From equation:

$(5)^2-4(1)(n)<0$

$25-4n<0$

$n>25/4$

So for the equation to have no real solutions:

$n>24/4=6.25$

We are choosing integers so n would have to be:

$7, 8 , 9, 10$

So for the probability of this we have:

$("favourable outcomes")/("all possible outcomes")=4/10=2/5$

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I choose a random integer $n$ $n$ between $1$ $1$ and $10$ $10$ inclusive. What is the probability that for the $n$ $n$ I chose, there exist no real solutions to the equation $x (x + 5) = - n$ $x(x+5) = -n$ ? Express your answer as a common fraction.

~ Question from AoPS ~

Background Information
Subject: Algebra
Focus: Percents
*Review Problem

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

I choose a random integer nnn between 111 and 101010 inclusive. What is the probability that for the nnn I chose, there exist no real solutions to the equation x(x+5) = -nx(x+5)=−nx(x+5) = -n? Express your answer as a common fraction.

~ Question from AoPS ~ Background Information Subject: Algebra Focus: Percents *Review Problem

1 Answer

Explanation:

Related questions

Impact of this question

I choose a random integer $n$ $n$ between $1$ $1$ and $10$ $10$ inclusive. What is the probability that for the $n$ $n$ I chose, there exist no real solutions to the equation $x (x + 5) = - n$ $x(x+5) = -n$ ? Express your answer as a common fraction.

~ Question from AoPS ~

Background Information
Subject: Algebra
Focus: Percents
*Review Problem