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Chapter 1: Problem 122

Find \(k\) so that the equation \(x^{2}-k x+4=0\) has a repeated real solution.

Short Answer

Expert verified
The values of \(k\) are 4 and -4.

Step by step solution

01

Understand the condition for repeated real solutions

A quadratic equation has repeated real solutions if and only if its discriminant is zero. The general form of a quadratic equation is given by: \[ax^2 + bx + c = 0\]The discriminant (螖) of this equation is calculated as: \[螖 = b^2 - 4ac\]
02

Identify the coefficients

Identify the coefficients from the given equation \(x^2 - kx + 4 = 0\). Here, \(a = 1\), \(b = -k\), and \(c = 4\).
03

Set the discriminant to zero

Set the discriminant to zero for there to be repeated real solutions: \[螖 = b^2 - 4ac = 0\]Substitute the identified values: \[( -k )^2 - 4 (1) (4) = 0\]
04

Solve for k

Solve the equation from Step 3: \[k^2 - 16 = 0\]Add 16 to both sides: \[k^2 = 16\]Take the square root of both sides: \[k = \pm 4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation
A quadratic equation is a polynomial equation of degree 2. It typically has the form:
  • \[ax^2 + bx + c = 0\]
Here, a, b, and c are coefficients. The coefficient a must be nonzero, otherwise the equation would be linear, not quadratic. Quadratic equations are fundamental in algebra and appear in various fields such as physics, engineering, and economics.
They usually have two solutions which can be real or complex, depending on the discriminant.
To solve a quadratic equation, you can use different methods like factoring, completing the square, or using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Discriminant
The discriminant of a quadratic equation provides crucial information about the nature of its solutions. For the quadratic equation \[ax^2 + bx + c = 0\], the discriminant (鈭�) is given by:
  • \[鈭� = b^2 - 4ac\]
The value of the discriminant determines the type of solutions the quadratic equation will have:
  • If \[鈭� > 0\], there are two distinct real solutions.
  • If \[鈭� = 0\], there is exactly one repeated real solution (or a real solution with multiplicity 2).
  • If \[鈭� < 0\], there are two complex solutions.
In the context of the given exercise, we set the discriminant to zero to find the value of k for which the quadratic equation \[x^2 - kx + 4 = 0\] has a repeated real solution. This condition simplifies the process and gives us the specific values to find the unique solution.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of x that satisfy \[ax^2 + bx + c = 0\]. Here are the methods you can use:
  • **Factoring**: This works when the quadratic can be factored into two binomial expressions.
  • **Completing the Square**: This transforms the quadratic into a perfect square trinomial which can then be solved for x.
  • **Quadratic Formula**: This method works for all quadratics and is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In the exercise, we used the discriminant to determine the condition for repeated real solutions. For the equation \[x^2 - kx + 4 = 0\], setting the discriminant to zero gives us the equation: \[k^2 - 16 = 0\]. Solving this, we find \[k = \pm 4\]. This means the quadratic equation will have a repeated real solution when k is either 4 or -4. These values ensure that the discriminant is zero, yielding one unique solution for the quadratic equation.

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