Question

If 81 is the discriminant of \(2 \mathrm{x}^{2}+5 \mathrm{x}-\mathrm{k}=0\) then the value of \(\mathrm{k}\) is (a) 5 (b) 7 (c) \(-7\) (d) 2

Short answer

The value of k is 7, which corresponds to option (b).

Step-by-step solution

Identify the coefficients of the quadratic equation

The given quadratic equation is \(2x^2 + 5x - k = 0\). The coefficients correspond to the equation form Ax^2 + Bx + C = 0, so we have A = 2, B = 5, and C= -k.

Write down the discriminant formula

The discriminant formula is D = B^2 - 4AC.

Substitute the given values into the discriminant formula

We are given that the discriminant (D) is 81, so we will substitute the given coefficients into the discriminant formula and set it equal to 81: \(81 = 5^2 - 4(2)(-k)\).

Solve the equation for k

Simplify the equation and solve for k: \[ \begin{aligned} 81 &= 25 + 8k\\ 56 &= 8k\\ k &= 7 \end{aligned} \] The value of k is 7, which corresponds to option (b).

Key concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). These equations are pivotal in various fields, including physics, engineering, and economics, due to their ability to model parabolic motion and other phenomena.

Understanding the structure of quadratic equations allows for the identification of their key properties, such as the axis of symmetry and the vertex of their graphical representation. The graph of a quadratic equation is a parabola that can open upwards or downwards depending on the sign of the coefficient \(a\). When solving these equations, it's crucial to determine whether there are two real solutions, one real solution, or no real solutions, which is related to the concept of the discriminant.

Quadratic Formula

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a universal method for solving quadratic equations, providing solutions for \(x\) by using the coefficients \(a\), \(b\), and \(c\) of the quadratic equation \(ax^2 + bx + c = 0\).

Why Use the Quadratic Formula?

• It gives exact solutions to a quadratic equation, even when other methods fail or are not readily apparent.
• It is applicable to any form of quadratic equation, regardless of complexity.
• The discriminant (\(b^2 - 4ac\)) within the formula helps to predict the nature of the roots without solving the entire equation.

The formula is derived by completing the square of a standard quadratic equation and is particularly useful when factoring by inspection is not feasible.

Solving Quadratic Equations

Solving quadratic equations entails finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). There are various methods to approach this task:

Factoring

If the quadratic can be factored into expressions such as \(a(x - r)(x - s) = 0\), then the solutions are \(x = r\) and \(x = s\).

Quadratic Formula

The quadratic formula offers solutions directly and is especially handy when the equation cannot be easily factored.

Completing the Square

This method involves converting the equation into a perfect square trinomial, allowing us to solve for \(x\) by taking the square root.

Graphing

Finding the x-intercepts of the parabola on its graph also yields the solutions to the equation.
Each method has its use cases, and understanding when to apply them is key to efficiently solving these equations. The discriminant within the quadratic formula is of particular importance, as it dictates whether solutions are real or complex, and if there are one or two solutions.