Use the discriminant to determine the following equations have that are two different rational solutions:two different irrational solutions or exactly one rational solution or two different imaginary solution.
x^2=5x+6
To determine the nature of the solutions of the equation x^2 = 5x + 6 using the discriminant, we need to apply the formula for the discriminant (D) which is given by D = b^2 - 4ac. In this equation, a is the coefficient of the x^2 term, b is the coefficient of the x term, and c is the constant term.
The given equation is: x^2 = 5x + 6
Comparing it to the standard form of a quadratic equation, which is ax^2 + bx + c = 0, we have:
a = 1 (coefficient of x^2)
b = -5 (coefficient of x)
c = -6 (constant term)
Substituting these values into the discriminant formula, we get:
D = (-5)^2 - 4(1)(-6)
D = 25 + 24
D = 49
Now, let's analyze the different possibilities based on the value of the discriminant:
1. Two different rational solutions:
If the discriminant (D) is a perfect square and is positive (D > 0), then the quadratic equation has two different rational solutions. In this case, D = 49, which is a perfect square and is positive.
2. Two different irrational solutions:
If the discriminant (D) is not a perfect square and is positive (D > 0), then the quadratic equation has two different irrational solutions. In this case, D = 49, which is a perfect square and is not a perfect square.
3. Exactly one rational solution:
If the discriminant (D) is zero (D = 0), then the quadratic equation has exactly one rational solution.
4. Two different imaginary solutions:
If the discriminant (D) is negative (D < 0), then the quadratic equation has two different imaginary solutions.
In the case of the equation x^2 = 5x + 6, since the discriminant is 49 (positive and a perfect square), the equation has two different rational solutions. To find these solutions, we can use the quadratic formula or factor the equation.