Question

Plato please help

Which statement describes the solution set of function f? f(x) = 3x^ 2 - 4x - 2

A- One real solution and one complex solution

B- Two complex solutions

C- Two real solutions

D- One real solution

Answer 1

Two real solutions

Explanation:

We can find the number and type of solutions in Quadratic Function/Equation by applying the discriminant. Discriminant is the formula that tells you the number and type of solutions, it’s derived from the Quadratic Formula. We know that the Quadratic Formula is:

`\displaystyle{x=(-b\pm √(b^2-4ac))/(2a)}`

The discriminant is inside the square root,
`\displaystyle{b^2-4ac}`. Discriminant tells the followings - if we evaluate discriminant and get:

• D > 0 (value greater than 0) then there are 2 real different solutions.
• D = 0 (value equals 0) then there are 2 real same solutions or in other word, there is only one real solution.
• D < 0 (value less than 0) then there are no real solutions, in other word, there are 2 complex solutions.

Step 1: Substitute a = 3, b = -4 and c = -2 in the discriminant formula

`\displaystyle{D=b^2-4ac \to (-4)^2-4(3)(-2)}`

Step 2: Evaluate the discriminant

Keep in mind that multiplying negative with negative always result in positives.

`\displaystyle{=(-4)(-4)-4(3)(-2)}\\\\\displaystyle{=16-12(-2)}\\\\\displaystyle{=16+24}\\\\\displaystyle{=40}`

Step 3: Consider the discriminant

Since the discriminant equals to 40, this satisfies the condition where D > 0 which says that there are 2 real different roots. Therefore, this function has 2 real solutions.

Please let me know if you have any questions!