f (x)= 2x square -3x-5

Given the function:

f(x) = 2x^2 - 3x - 5

a = 2

b = -3

c = -5

Determine the roots.

Solution:

In order to solve for the roots of a function, 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 -5. 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(2)(-5))/2(2)
x = 3 ± √(9 + 40)/4
x = 3 ± √(49)/4
x = 3 ± 7/4

Solve for + - separately.

x = 3 + 7 / 4
x = 10/4
x =  2.5

x = 3 - 7/4
x = -4/4
x = -1

Final answer:

x =  2.5
x = -1