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QuestionIdentify the type of roots for the function f(x)=−x2−3x+1f(x)=-x^{2}-3x+1. Choose from: a. 1 rational, b. 2 rational, c. 2 irrational, d. 2 imaginary.

Studdy Solution

STEP 1

Assumptions1. The function is a quadratic function of the form f[x]=ax+bx+cf[x]=ax^{}+bx+c where a, b and c are constants. . The roots of the function can be found using the quadratic formula x=−b±b−4acax=\frac{-b \pm \sqrt{b^{}-4ac}}{a}3. The discriminant, =b−4ac = b^{}-4ac, determines the nature of the roots.

STEP 2

First, we need to identify the coefficients a, b, and c in the given quadratic function.
In the given function f[x]=−x2−x+1f[x]=-x^{2}-x+1, we havea = -1, b = -, c =1

STEP 3

Next, we calculate the discriminant D using the formula =b2−ac = b^{2}-ac

STEP 4

Plug in the values for a, b, and c into the discriminant formula.
=(−3)2−4(−1)(1) = (-3)^{2}-4(-1)(1)

STEP 5

Calculate the value of the discriminant.
=9−(−4)=13 =9-(-4) =13

STEP 6

Now, we analyze the value of the discriminant to determine the nature of the roots.
If D >0, the roots are real and distinct. If D =0, the roots are real and equal. If D <0, the roots are complex or imaginary.
In our case, D =13 which is greater than0. So, the roots are real and distinct.

STEP 7

Finally, we determine whether the roots are rational or irrational.
A root is rational if the discriminant is a perfect square, and irrational if the discriminant is not a perfect square.
In our case, D =13 which is not a perfect square. So, the roots are irrational.
Therefore, the correct answer isc.2 distinct irrational roots.

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