help me please im confused
Consider the quadratic expression X^2 + 4x + c = 0.
For what range of values does the equation have two complex roots?
what am i doing here can someone show me step by step so i can at least try to do the other problems i have
Hi. I answered you previous post and it was almost similar, except that here, the roots must be complex/imaginary so D must be less than zero. Anyway,
Recall the formula for discriminant. For a quadratic equation in the general form, ax^2 + bx + c = 0,
D = b^2 - 4ac
if
D = 0 : real, equal/double root
D > 0 : two real, unequal roots
D < 0 : two imaginary roots
Since we're required to have complex/imaginary, D < 0, and solve for the unknown, c.
x^2 + 4x + c = 0
a = 1
b = 4
c = ?
Substituting to the discriminant formula, (D < 0)
0 < 4^2 - 4*1*c
0 < 16 - 4c
4c < 16
c < 4
Hope this helps~ :3
I disagree that the answer is c < 4. The correct answer is c > 4.
Lets see the steps again:
x^2 + 4x + c = 0, b^2 - 4ac < 0
4^2 - 4(1)(c) < 0
16 - 4c < 0
16 < 4c
4c > 16
Therefore c > 4
Well, well, well, looks like we have a quadratic equation dilemma! Don't worry, I'm here to put a smile on that confused face of yours. Let's break it down step by step so you can solve this equation and tackle those other problems successfully.
To determine the range of values for which the equation X^2 + 4x + c = 0 has two complex roots, we need to analyze the discriminant.
Now, what's a discriminant, you ask? It's not a fancy judge at a beauty pageant, no. The discriminant of a quadratic equation is found by taking the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation Ax^2 + Bx + C = 0.
In this case, the quadratic equation is X^2 + 4x + c = 0. So, our coefficients are: a = 1, b = 4, and c = c.
Now, let's dig into the discriminant formula for this equation:
Discriminant = b^2 - 4ac
= (4)^2 - 4(1)(c)
= 16 - 4c
To have two complex roots (or imaginary roots), the discriminant must be less than zero. Why? Because you know what they say, imaginary numbers need a bit of negativity in their lives!
Therefore, we want the discriminant to satisfy the condition:
16 - 4c < 0
Now, simple math time! Let's solve this baby:
16 < 4c
4 < c
So, the range of values for which the equation has two complex roots is when c is greater than 4. In other words, c > 4.
Voila! Now you're armed with the knowledge to solve this problem like a boss. Good luck with your other problems, and remember, a problem is just an opportunity for you to show off your incredible quadratic-solving skills!
To determine the range of values for which the equation has two complex roots, we can consider the discriminant of the quadratic expression. The discriminant is given by the formula:
Discriminant = b^2 - 4ac
For the quadratic expression X^2 + 4x + c = 0, the coefficient of the x^2 term, a, is 1, the coefficient of the x term, b, is 4, and the constant term, c, is c.
Step 1: Substitute the values into the formula for the discriminant.
Discriminant = (4^2) - 4(1)(c)
= 16 - 4c
Step 2: Determine the conditions for two complex roots.
For the quadratic equation to have two complex roots, the discriminant should be less than 0 (negative).
Step 3: Set up the inequality.
Discriminant < 0
Step 4: Solve the inequality for c.
16 - 4c < 0
Step 5: Simplify and solve for c.
-4c < -16
c > 4
The expression X^2 + 4x + c = 0 has two complex roots for values of c greater than 4.
I hope this explanation helps you understand how to approach this type of problem. Let me know if you have any further questions!
Of course, I'll be happy to help you understand how to solve this problem step by step.
To determine the range of values for which the quadratic equation has two complex roots, we need to analyze the discriminant (denoted as Δ) of the quadratic equation:
Δ = b^2 - 4ac
In our equation, X^2 + 4x + c = 0, the coefficient of x is 4, and the coefficient of x^2 is 1, while c represents a constant term.
For the equation to have two complex roots, the discriminant Δ must be negative because the square root of a negative number gives a complex number.
Step 1: Write down the general form of the discriminant formula:
Δ = b^2 - 4ac
Step 2: Substitute the values from the given quadratic equation into the formula:
Δ = (4)^2 - 4(1)(c)
Step 3: Simplify the equation:
Δ = 16 - 4c
Step 4: Set the discriminant Δ to be less than 0:
Δ < 0
Step 5: Solve the inequality for c:
16 - 4c < 0
Step 6: Solve for c:
16 < 4c
4 < c
Therefore, the range of values for c in which the quadratic equation has two complex roots is c > 4. In other words, any value of c greater than 4 will result in the equation having two complex roots.
I hope that helps! Let me know if you have any further questions or if there's anything else I can assist you with.