Use the value of the discriminant to determine the number and type of roots for the equation: (x^2+20=12x-16)
1 real, irrational
2 real, rational
no real
1 real, rational
For quadratics, either both roots are real or both are complex. In a special case where the discriminant is zero, there is one root.
In your case, the equation should be written in the ax^2 + bx + c = 0 form, which would be
x^2 -12x + 36 = 0.
discriminant = b^2 - 4ac = 144-144 = 0
So there is only one root.
x = -b/2a = 6, which is a rational number
To determine the number and type of roots for the equation (x^2+20=12x-16), we need to first rearrange the equation into the standard quadratic form, which is ax^2 + bx + c = 0.
The given equation can be rearranged as follows:
x^2 - 12x + 36 = 0
Now we can identify the values of a, b, and c:
a = 1
b = -12
c = 36
Next, we can calculate the discriminant using the formula:
D = b^2 - 4ac
Substituting in the values we found:
D = (-12)^2 - 4(1)(36)
D = 144 - 144
D = 0
The discriminant is equal to zero, which means that there is only one real root for the equation. Additionally, since the discriminant is not negative, the root will be rational.
Therefore, the answer is 1 real, rational root.
To determine the number and type of roots for an equation, we need to consider the value of the discriminant. The discriminant is a mathematical expression derived from the coefficients of the equation.
For a quadratic equation in the form ax^2 + bx + c = 0, the discriminant (denoted as Δ) is given by the formula: Δ = b^2 - 4ac.
In the given equation x^2 + 20 = 12x - 16, we need to rewrite it in the standard quadratic form by subtracting 12x and adding 16 to both sides:
x^2 - 12x + 36 = 0.
Comparing this equation with the standard quadratic form, we can see that a = 1, b = -12, and c = 36.
Now, let's calculate the discriminant by substituting the values in the formula:
Δ = (-12)^2 - 4(1)(36)
= 144 - 144
= 0.
The discriminant Δ is equal to 0.
Based on the value of the discriminant, we can determine the number and type of roots:
1. If Δ > 0, then the quadratic equation has two real roots.
2. If Δ = 0, then the quadratic equation has one real root.
3. If Δ < 0, then the quadratic equation has no real roots.
Since the discriminant in the given equation is equal to 0, we can conclude that it has one real root.
Therefore, the correct option is: 1 real, rational.