Question

how many real solutions does the equation have?
(n^2 - 75 = -86)
no real solution
one real solution
two real solutions

Explanation:

Step1: Rewrite the equation

First, we rewrite the equation \( n^{2}-75 = -86 \) to the standard quadratic form \( ax^{2}+bx + c = 0 \). By adding 86 to both sides, we get \( n^{2}+ 11=0 \), where \( a = 1 \), \( b = 0 \), and \( c = 11 \).

Step2: Use the discriminant formula

The discriminant of a quadratic equation \( ax^{2}+bx + c = 0 \) is given by \( D=b^{2}-4ac \). Substituting the values of \( a \), \( b \), and \( c \) into the formula, we have \( D = 0^{2}-4\times1\times11 \).

Step3: Calculate the discriminant

Calculating the discriminant: \( D=0 - 44=- 44 \).

Step4: Analyze the discriminant

Since the discriminant \( D=-44<0 \), a quadratic equation has no real solutions when the discriminant is negative.

Answer:

no real solution