Answered

Compute the discriminant. Then determine the number and type of solutions of the given equation:



$x^2-7 x-4=0$



What is the discriminant?

Verified Answer :

  • Identify coefficients: \(a=1, b=-7, c=-4\).
  • Apply the discriminant formula: \(D = b^2 - 4ac\).
  • Calculate: \(D = (-7)^2 - 4(1)(-4) = 49 + 16 = 65\).
  • Conclusion: Since \(D > 0\), the equation has two distinct real solutions.
  • The discriminant is \(\boxed{65}\).

Explanation

  1. Identifying Coefficients and the Discriminant Formula

To analyze the quadratic equation \(x^2 - 7x - 4 = 0\), we first identify its coefficients, which are in the standard form \(ax^2 + bx + c = 0\). Comparing our equation to this form, we find the coefficients to be \(a = 1\), \(b = -7\), and \(c = -4\). The discriminant, denoted by \(D\), is a key parameter that reveals the nature of the equation's roots and is calculated using the formula \(D = b^2 - 4ac\).

  1. Calculating the Discriminant

We substitute the identified coefficients into the discriminant formula:
\[D = (-7)^2 - 4(1)(-4).\]
Calculating the square of \(-7\), we get \(49\). Next, we calculate the product \(-4 \times 1 \times -4\), which is \(16\). Adding these results, we find:
\[D = 49 + 16 = 65.\]
The discriminant of the equation is therefore \(65\).

  1. Determining the Nature of Solutions

The nature of the solutions for a quadratic equation is determined by the sign of the discriminant:

  • If \(D > 0\), the equation has two distinct real solutions.
  • If \(D = 0\), the equation has exactly one real solution.
  • If \(D < 0\), the equation has two complex solutions.

Since our discriminant \(D = 65\) is greater than \(0\), we conclude that the equation \(x^2 - 7x - 4 = 0\) has two distinct real solutions.

Examples

Quadratic equations and their discriminants are essential in many fields, such as physics and engineering, to determine if a system will reach a certain state. For example, when launching a projectile, the quadratic equation for its height over time might have a discriminant that tells you if it will hit the ground (real roots) or if the model needs adjustment (no real roots at that time). Calculating the discriminant allows scientists to predict the behavior of moving objects efficiently.

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