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If \[81\] is the discriminant of \[2{x^2} + 5x - k = 0\], then the value of \[k\] is
A. \[5\]
B. \[7\]
C. \[ - 7\]
D. \[2\]

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Last updated date: 17th Jul 2024
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Answer
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Hint: We know that to solve the given equation we need to use the discriminant formula and then substitute the values in the given equation to get the desired result. The discriminant is a factor that helps to find the exact roots of a quadratic equation if the equation is not a perfect square.

Formula Used: We have used the formula of discriminant that is given below
 \[\Delta = {b^2} - 4ac\]

Complete step-by-step solution:
We are given an equation that is \[2{x^2} + 5x - k = 0\]
Now, compare the given equation to the standard quadratic equation \[a{x^2} + bx + c = 0\] where the values of a, b and c are 2, 5 and -k, respectively.
Now we apply the formula of discriminant in the given equation, and we get
\[
  {b^2} - 4ac = 81 \\
   \Rightarrow {\left( 5 \right)^2} - 4 \times 2 \times \left( { - k} \right) = 81 \\
   \Rightarrow 25 + 8k = 81 \\
   \Rightarrow 8k = 81 - 25
 \]
Further Simplifying, we get,
\[
  8k = 56 \\
   \Rightarrow k = 7
 \]

Hence, the value of k is 7, so, option B is correct.

Additional information: A discriminant is a term contained within a radical symbol (square root) of the quadratic formula. In mathematics, the discriminant is used to determine the nature of the roots of a quadratic equation. The discriminant value determines whether the roots of the quadratic equation are real or imaginary, equal or unequal. Similarly, for higher degree polynomials, the discriminant is always a polynomial function of the coefficients.

Note: Many students made miscalculations while substituting the wrong values of a, b, and c in the formula of discriminant so make sure about the formula and compare the values according to the signs of \[a{x^2} + bx + c = 0\] and also solve the question with the help of the formula.