use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solutions; exactly one rational solution; or two different imaginery solutions. Help please.
I for got to add this part in 8x^2+7x+3=0
the discriminant is b^2 - 4ac
d = 49 - 96 = -47 < 0
so, there are no real roots: two complex roots
To determine the nature of the solutions of a quadratic equation, we can use the discriminant, which is the expression under the square root in the quadratic formula. The discriminant can be calculated using the formula:
Discriminant (D) = b^2 - 4ac
Here's how the discriminant can help classify the solutions:
1. Two different rational solutions: If the discriminant (D) is a perfect square and greater than zero (D > 0), then the equation will have two different rational solutions. Both solutions will be real numbers that can be expressed as fractions.
2. Two different irrational solutions: If the discriminant (D) is not a perfect square and greater than zero (D > 0), then the equation will have two different irrational solutions. Both solutions will be real numbers that cannot be expressed as fractions.
3. Exactly one rational solution: If the discriminant (D) is zero (D = 0), then the equation will have exactly one rational solution. The solution will be a real number that can be expressed as a fraction.
4. Two different imaginary solutions: If the discriminant (D) is negative (D < 0), then the equation will have two different imaginary solutions. The solutions will involve the imaginary unit, typically represented as "i."
To determine the nature of the solutions, calculate the discriminant and apply the above rules.