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If a, b, c in R and the quadratic eq...

If ` a, b, c in R ` and the quadratic equation ` x^2 + (a + b) x + c = 0 ` has no real roots then

A

`c(a+b+c) gt 0`

B

`c+(a+b+c)c gt 0`

C

`c-c(a+b+c) gt 0`

D

`c(a+b-c) gt 0`

Text Solution

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The correct Answer is:
To determine the condition under which the quadratic equation \( x^2 + (a + b)x + c = 0 \) has no real roots, we need to analyze the properties of the quadratic function. ### Step-by-Step Solution: 1. **Understanding the Condition for No Real Roots**: A quadratic equation \( ax^2 + bx + c = 0 \) has no real roots if its discriminant \( D < 0 \). The discriminant for our equation is given by: \[ D = (a + b)^2 - 4c \] For the quadratic to have no real roots, we require: \[ (a + b)^2 - 4c < 0 \] 2. **Rearranging the Inequality**: We can rearrange the inequality to isolate \( c \): \[ (a + b)^2 < 4c \] This means that \( c \) must be greater than one-fourth of the square of the sum \( (a + b) \). 3. **Conclusion**: Therefore, the condition for the quadratic equation \( x^2 + (a + b)x + c = 0 \) to have no real roots is: \[ c > \frac{(a + b)^2}{4} \]
To determine the condition under which the quadratic equation \( x^2 + (a + b)x + c = 0 \) has no real roots, we need to analyze the properties of the quadratic function. ### Step-by-Step Solution: 1. **Understanding the Condition for No Real Roots**: A quadratic equation \( ax^2 + bx + c = 0 \) has no real roots if its discriminant \( D < 0 \). The discriminant for our equation is given by: \[ D = (a + b)^2 - 4c ...
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