Problem 108 - Algebra for College Students

Question

Solve \(x^{2}-4 x+k=0\) for \(k=0,4,5,\) and 10. a) When does the equation have only one solution? b) For what values of \(k\) are the solutions real? c) For what values of \(k\) are the solutions imaginary?

Step-by-Step Solution

Answer

a) k = 4b) k = 0, 4c) k = 5, 10

Step 1 - Determine the Discriminant

For a quadratic equation of the form x^2 + bx + c = 0the discriminant D = b^2 - 4ac.Determine the discriminant for the given equation.

Step 2 - Calculate the Discriminant for Each k Value

Substitute the values of k (0, 4, 5, 10) into the equation and calculate the discriminant D.For k = 0: D = (-4)^2 - 4(1)(0) = 16For k = 4: D = (-4)^2 - 4(1)(4) = 0For k = 5: D = (-4)^2 - 4(1)(5) = -4For k = 10: D = (-4)^2 - 4(1)(10) = -24

Step 3 - Analyze the Discriminant Values

Use the discriminant to determine the nature of the solutions.If D > 0, there are two distinct real solutions.If D = 0, there is exactly one real solution.If D < 0, the solutions are imaginary.

Step 4 - Answer When Equation Has Only One Solution

The equation has only one solution when the discriminant is zero.When k = 4, D = 0, so the equation has exactly one solution for k = 4.

Step 5 - Answer for Real Solutions

The solutions are real when the discriminant is greater than or equal to zero.For k = 0: D = 16 (real solutions)For k = 4: D = 0 (one real solution)

Step 6 - Answer for Imaginary Solutions

The solutions are imaginary when the discriminant is less than zero.For k = 5: D = -4 (imaginary solutions)For k = 10: D = -24 (imaginary solutions)

Key Concepts

discriminant in quadratic equations

In quadratic equations, the discriminant is a crucial tool for understanding the nature of the solutions. For a general quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by the formula: \[\text{D} = b^2 - 4ac\]. The value of the discriminant tells us whether the solutions to the equation are real or imaginary, and whether they are distinct or repeated.

real solutions

Real solutions in quadratic equations occur when the discriminant is greater than or equal to zero.
We use the following rules to understand this:

• If \[\text{D} > 0\], there are two distinct real solutions.
• If \[\text{D} = 0\], there is exactly one real solution, which is also called a repeated root.

For example, in the given equation \(x^2 - 4x + k = 0\):
For \k = 0\, \D = 16\ (2 real solutions)
For \k = 4\, \D = 0\ (1 real solution)

imaginary solutions

Imaginary solutions occur when the discriminant is less than zero, indicating that the quadratic equation has no real solutions. Instead, it results in complex numbers.
In mathematical terms:

• If \[\text{D} < 0\], the solutions are imaginary.

For our equation \(x^2 - 4x + k = 0\):
For \k = 5\, \D = -4\ (2 imaginary solutions)
For \k = 10\, \D = -24\ (2 imaginary solutions)

quadratic discriminant analysis

Quadratic discriminant analysis involves examining the discriminant to determine the nature and number of solutions of a quadratic equation. It allows us to see if the solutions are real or imaginary, and whether the real solutions are distinct or repeated.
Let's break down the discriminant analysis for the equation \x^2 - 4x + k = 0\ based on various \text{k}\ values:

• \k = 0\rightarrow \[D = 16\] (2 real solutions)
• \k = 4\rightarrow \[D = 0\] (1 real solution)
• \k = 5\rightarrow \[D = -4\] (2 imaginary solutions)
• \k = 10\rightarrow \[D = -24\] (2 imaginary solutions)

Thus, quadratic discriminant analysis provides a systematic way to comprehend different outcomes of quadratic equations based on their coefficients.