Find a negative real number such that the square of the sum of the number and 3 is equal to 25
(n+3)^2 = 25
n+3 = -5
n = -8
To find a negative real number that satisfies the given equation, we can use algebraic equations and solve for the unknown number.
Let's proceed step by step:
1. Assume the unknown number to be x (a negative real number).
2. The sum of the number and 3 is (x + 3).
3. The square of the sum can be expressed as (x + 3)^2.
4. According to the problem statement, the square of the sum is equal to 25. Therefore, we have the equation: (x + 3)^2 = 25.
5. Expand the squared term: (x + 3)(x + 3) = 25.
6. Apply the FOIL method: x^2 + 6x + 9 = 25.
7. Arrange the equation in standard form: x^2 + 6x + 9 - 25 = 0.
8. Simplify: x^2 + 6x - 16 = 0.
Now we have a quadratic equation in standard form. To solve it, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients from the quadratic equation (ax^2 + bx + c = 0).
In our case, a = 1, b = 6, and c = -16.
9. Substitute the values into the quadratic formula:
x = (-6 ± √(6^2 - 4(1)(-16))) / (2(1)).
10. Simplify:
x = (-6 ± √(36 + 64)) / 2.
x = (-6 ± √100) / 2.
11. Further simplify:
x = (-6 ± 10) / 2.
x = (4/2) or (-16/2).
x = 2 or -8.
Since we are looking for a negative real number, the solution is x = -8.
Therefore, a negative real number, such that the square of the sum of the number and 3 is equal to 25, is x = -8.