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Magazine Chapter 1: Problem 74
Find all values of \(k\) that ensure that the given equation has exactly one solution. \(k x^{2}+36 x+k=0\)
Short Answer
Expert verified
The values of \(k\) are 18 and -18.
Step by step solution
01
Identify the form of the equation
The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = k\), \(b = 36\), and \(c = k\).
02
Condition for one solution
A quadratic equation has exactly one solution when its discriminant equals zero. The discriminant \(\Delta\) of the equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). To find when it has exactly one solution, set \(\Delta = 0\).
03
Substitute into the discriminant formula
Since \(b = 36\), \(a = k\), and \(c = k\), substitute these into the discriminant formula: \(\Delta = 36^2 - 4(k)(k)\).
04
Simplify the discriminant equation
Simplify the equation: \(\Delta = 1296 - 4k^2\).
05
Set the discriminant to zero
For the equation to have exactly one solution, set the discriminant to zero: \(1296 - 4k^2 = 0\).
06
Solve for \(k\)
Rearrange the equation: \(4k^2 = 1296\). Divide both sides by 4 to get \(k^2 = 324\).
07
Find the values of \(k\)
Take the square root of both sides to find \(k\): \(k = \pm 18\).
08
Conclusion
The values of \(k\) that ensure the equation has exactly one solution are \(k = 18\) and \(k = -18\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Discriminant
The discriminant is a crucial part of understanding and solving quadratic equations. In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is defined as \(b^2 - 4ac\). The discriminant helps determine the nature of the solutions for the equation.
- Positive Discriminant: If \(\Delta > 0\), the quadratic equation has two distinct real solutions.
- Zero Discriminant: If \(\Delta = 0\), there is exactly one unique solution because the equation has a repeated root.
- Negative Discriminant: If \(\Delta < 0\), the equation has no real solutions; instead, it has two complex conjugate solutions.
One Solution
When a quadratic equation has exactly one solution, it means the parabola defined by the equation \(ax^2 + bx + c = 0\) touches the x-axis at exactly one point. This situation is also known as having a double root or a repeated root.
The condition for a quadratic to have exactly one solution is met when its discriminant is zero. This simplifies to:\[\Delta = b^2 - 4ac = 0\]Using the given equation from the problem, \(kx^2 + 36x + k = 0\), the substitution into the discriminant gives:\[36^2 - 4(k)(k) = 0\]Solving this, we find that \(k^2 = 324\). Thus, \(k\) could be either 18 or -18 for the equation to hold true with exactly one solution.
The condition for a quadratic to have exactly one solution is met when its discriminant is zero. This simplifies to:\[\Delta = b^2 - 4ac = 0\]Using the given equation from the problem, \(kx^2 + 36x + k = 0\), the substitution into the discriminant gives:\[36^2 - 4(k)(k) = 0\]Solving this, we find that \(k^2 = 324\). Thus, \(k\) could be either 18 or -18 for the equation to hold true with exactly one solution.
Solving Quadratic Equations
Solving quadratic equations involves finding the x-values that satisfy the equation \(ax^2 + bx + c = 0\). There are several methods to solve quadratic equations:
- Factoring: This method involves rewriting the equation so that it can be expressed as a product of two binomial expressions.
- Completing the Square: This method makes the quadratic expression into a perfect square trinomial.
- Quadratic Formula: This universal method uses the formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find the roots directly.
