Consider the equation:
4x^2 – 16x + 25 = 0
(a) Show how to compute the discriminant, b^2 – 4ac, and then state whether there is one real-number solution, two different real-number solutions, or two different imaginary-number solutions.
(b) Use the quadratic formula to find the exact solutions of the equation. Show work. Simplify the final results as much as possible.
What is your question. The problem is rather self-explanatory.
This is my first time doing a problem like this and need some guidence. I have 5 more problems like this one and would like some help so I know how to do the rest. Thanks!
The page will give you a good head-start.
http://www.mathnstuff.com/math/spoken/here/2class/320/quadequ.htm
To determine the nature and exact solutions of the equation 4x^2 – 16x + 25 = 0, we need to compute the discriminant and then use the quadratic formula.
(a) Computing the discriminant:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 – 4ac. In this equation, a = 4, b = -16, and c = 25.
Substituting these values into the formula, we have:
Discriminant = (-16)^2 - 4 * 4 * 25
= 256 - 400
= -144
The discriminant of -144 tells us that the quadratic equation has two different imaginary-number solutions.
(b) Using the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values from the equation, we have:
x = (-(-16) ± √((-16)^2 - 4 * 4 * 25)) / (2 * 4)
= (16 ± √(256 - 400)) / 8
= (16 ± √(-144)) / 8
Since the discriminant is negative, we have the square root of a negative number (√(-144)) which results in an imaginary number. We simplify the radical as follows:
√(-144) = √(144 * -1) = √(144) * √(-1) = 12i
So, the solutions to the equation are:
x₁ = (16 + 12i) / 8
x₂ = (16 - 12i) / 8
To simplify further, we can divide the numerator by 4, which gives us:
x₁ = (4 + 3i) / 2
x₂ = (4 - 3i) / 2
Therefore, the exact solutions to the equation 4x^2 – 16x + 25 = 0 are x₁ = (4 + 3i)/2 and x₂ = (4 - 3i)/2.