Questions: In Exercises 19-26, find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. 21. 4 n^2 - 4 n - 24 = 0 23. 4 x^2 = 5 x - 10

In Exercises 19-26, find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. 21. 4 n^2 - 4 n - 24 = 0 23. 4 x^2 = 5 x - 10
Transcript text: In Exercises 19-26, find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. 21. $4 n^{2}-4 n-24=0$ 23. $4 x^{2}=5 x-10$
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Solution

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Solution Steps

To find the discriminant of a quadratic equation in the form \( ax^2 + bx + c = 0 \), we use the formula \( b^2 - 4ac \). The discriminant helps us determine the number and type of solutions: if it's positive, there are two distinct real solutions; if it's zero, there is one real solution; and if it's negative, there are two complex solutions.

Step 1: Calculate the Discriminant for Equation 21

For the quadratic equation \( 4n^2 - 4n - 24 = 0 \), we have:

  • \( a = 4 \)
  • \( b = -4 \)
  • \( c = -24 \)

The discriminant \( D \) is calculated as follows: \[ D = b^2 - 4ac = (-4)^2 - 4 \cdot 4 \cdot (-24) = 16 + 384 = 400 \]

Since \( D = 400 > 0 \), there are two distinct real solutions.

Step 2: Calculate the Discriminant for Equation 23

For the quadratic equation \( 4x^2 - 5x + 10 = 0 \), we have:

  • \( a = 4 \)
  • \( b = -5 \)
  • \( c = 10 \)

The discriminant \( D \) is calculated as follows: \[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 4 \cdot 10 = 25 - 160 = -135 \]

Since \( D = -135 < 0 \), there are two complex solutions.

Final Answer

For equation 21, there are two distinct real solutions. For equation 23, there are two complex solutions.

Thus, the final answers are:

  • For equation 21: \\(\boxed{2 \text{ distinct real solutions}}\\)
  • For equation 23: \\(\boxed{2 \text{ complex solutions}}\\)
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