the sum of anon-negative number and its square is less than 12. Whate could the number be?
x + x^2 < 12
x^2 + x - 12 < 0
(x+4)(x-3) < 0
but x > 0
so 0 < x < 3
that is, x has to be between 0 and 3
-× ^2 less than 12
X^2greater than 12
To find the possible values of the non-negative number, we can set up an inequality based on the given information.
Let's assume the non-negative number is "x".
According to the problem, the sum of the non-negative number (x) and its square (x^2) is less than 12:
x + x^2 < 12
Next, we can rearrange the inequality to a quadratic equation form:
x^2 + x - 12 < 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. However, since we are looking for non-negative values, we can focus on finding the solutions within the given constraints.
To factor the quadratic equation, we need to find two numbers whose product is -12 and whose sum is +1 (coefficient of the linear term):
Since -3 * 4 = -12 and -3 + 4 = 1, we can factor the quadratic equation:
(x - 3)(x + 4) < 0
Now, we need to determine the sign of the inequality for different intervals to find the valid values of x.
Considering the intervals:
1. x < -4: In this interval, both factors (x - 3) and (x + 4) are negative. The product of two negatives is positive. Therefore, the inequality is not satisfied.
2. -4 < x < 3: In this interval, the first factor (x - 3) is negative, while the second factor (x + 4) is positive. The product of a negative and a positive is negative. Hence, this interval satisfies the inequality.
3. x > 3: In this interval, both factors (x - 3) and (x + 4) are positive. The product of two positives is positive. Thus, the inequality is not satisfied.
So, the valid interval for x is -4 < x < 3.
Therefore, the non-negative number that satisfies the given conditions can be any number greater than -4 and less than 3. Note that negative numbers are not valid since the problem only allows non-negative numbers.