The equation given in the question was 5x^2 - (k+2)x + 7k + 1. So, first of all, let me clarify that 5x^2 - (k+2)x + 7k + 1 is an expression, and not an equation. 5x^2 - (k+2)x + 7k + 1 = 0 will be called an equation, as it is "something equals 0". This is one common mistake in mathematics, and should be avoided.
(1) We have the equation 5x² - (k+2)x + 7k + 1 = 0.
Given that one root of the equation is 3. Which means x = 3 will satisfy the equation.
=> 5(3)² - (k+2)(3) + 7k + 1 = 0
=> 5(9) – (k+2)3 + 7k + 1 = 0
=> 45 – 3k - 6 + 7k + 1 = 0
=> 40 + 4k = 0
=> 4k = -40
=> k = -40/4 = -10
Hence, the value of k is -10 if 3 is a root of the equation.
(2) Given that one root is negative of the other root.
Now, consider this situation very carefully. You have two numbers, one number is negative of the other (say, a and –a). It can be easily understood that the sum of these numbers will be 0, like (–a) + (a) = 0.
So, in this question, as we know that one root is negative of the other, the sum of the roots will be 0.
We know that sum of roots of a Quadratic Equation ax² + bx + c = 0 is -b/a
Here, we'll have
- [-(k + 2)]/5 = 0
=> (k + 2)/5 = 0
=> (k + 2) = 0
=> k = -2
Hence, the value of k is -2 if one root of the equation is negative of the other.
(1) We have the equation 5x² - (k+2)x + 7k + 1 = 0.
Given that one root of the equation is 3. Which means x = 3 will satisfy the equation.
=> 5(3)² - (k+2)(3) + 7k + 1 = 0
=> 5(9) – (k+2)3 + 7k + 1 = 0
=> 45 – 3k - 6 + 7k + 1 = 0
=> 40 + 4k = 0
=> 4k = -40
=> k = -40/4 = -10
Hence, the value of k is -10 if 3 is a root of the equation.
(2) Given that one root is negative of the other root.
Now, consider this situation very carefully. You have two numbers, one number is negative of the other (say, a and –a). It can be easily understood that the sum of these numbers will be 0, like (–a) + (a) = 0.
So, in this question, as we know that one root is negative of the other, the sum of the roots will be 0.
We know that sum of roots of a Quadratic Equation ax² + bx + c = 0 is -b/a
Here, we'll have
- [-(k + 2)]/5 = 0
=> (k + 2)/5 = 0
=> (k + 2) = 0
=> k = -2
Hence, the value of k is -2 if one root of the equation is negative of the other.
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