What are the exact solutions of x2 − 3x − 1 = 0? Select one: a. x = the quantity of 3 plus or minus the square root of 5 all over 2 Incorrect b. x = the quantity of negative 3 plus or minus the square root of 5 all over 2 c. x = the quantity of 3 plus or minus the square root of 13 all over 2 d. x = the quantity of negative 3 plus or minus the square root of 13 all over 2

Given the quadratic function:

f(x) = x^2 - 3x - 1

a = 1

b = -3

c= -1

Determine the value of x

Solution:
In order to solve for the roots or zeros of an equation, we simply must look at the 2nd and 3rd value. First we must think of two numbers that when added, the answer is -3, and when multiplied, the answer is -1. Hence, if we are unable to find the number, we will use the quadratic formula.

The Quadratic formula:

x = −b ± √(b^2 − 4ac)/2a

is used to solve quadratic equations where a ≠ 0, in the form
ax^2+bx+c=0

When b^2−4ac=0 there is one real root.

When b^2−4ac>0 there are two real roots.

When b^2−4ac<0 there are no real roots, only a complex number.

Substitute the given values of a, b and c to the quadratic formula.

x = −b ± √(b^2 − 4ac)/2a

x = −(-3) ± √((-3)^2 − 4(1)(-3))/2(1)

x = 3 ± √(9 + 12)/2

x = 3 ± √(21)/2

x = 3 ± 4.5/2

Solve for +- separately.

x = 3 + 4.5/2
x = 7.5/2
x = 3.75

x = 3 - 4.5/2
x = -1.5/2
x = -0.75

Final answer:

x = 3.75

x = -0.75