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Magazine Chapter 11: Problem 69
One solution of \(k x^{2}+3 x-k=0\) is \(-2 .\) Find the other.
Short Answer
Expert verified
\(x = \frac{1}{2}\)
Step by step solution
01
Substitute the Known Solution
Given that one solution of the equation is \(-2\), substitute \(x = -2\) into the quadratic equation \(k x^{2}+3 x-k=0\). This gives: \(k(-2)^{2} + 3(-2) - k = 0\)
02
Simplify the Equation
Simplify the substituted equation: \(4k - 6 - k = 0\). Combine like terms: \(3k - 6 = 0\)
03
Solve for \(k\)
Solve for \(k\) by isolating \(k\): \(3k = 6\), then \(k = 2\)
04
Write the New Quadratic Equation
Substitute \(k = 2\) back into the original quadratic equation to get \(2x^{2}+3x-2=0\)
05
Use the Quadratic Formula
To find the other solution of \(2x^{2}+3x-2=0\), use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a = 2\), \(b = 3\) and \(c = -2\)
06
Calculate the Discriminant
Calculate the discriminant: \(b^{2} - 4ac = 3^{2} - 4(2)(-2) = 9 + 16 = 25\)
07
Find the Solutions
Substitute the discriminant into the quadratic formula: \(x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}\). This gives two solutions: \(x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}\) and \(x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2\)
08
Identify the Other Solution
Since \(x = -2\) is already known, the other solution is \(x = \frac{1}{2}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substituting Known Solutions
When you know one of the solutions to a quadratic equation, you can use this information to simplify and solve the equation. In the given problem, we know that one root of the quadratic equation \( kx^2 + 3x - k = 0 \) is \(x = -2\). By substituting \(x = -2\) into the equation, we can find the constant coefficient \(k\). This step transforms our original quadratic equation into something more manageable. When you replace \(x\) with \(-2\), the equation becomes: \(k(-2)^2 + 3(-2) - k = 0\).
Simplifying Equations
Simplifying the equation is crucial to make it easier to solve. Once we have substituted \(-2\) for \(x\), we simplify the equation step-by-step.
We first calculate \((-2)^2 = 4\). This gives us: \(4k - 6 - k = 0\). Next, we combine like terms \(4k - k\) to get \(3k\). This reduction transforms our equation into a much simpler form: \(3k - 6 = 0\). Solving for \(k\):
We first calculate \((-2)^2 = 4\). This gives us: \(4k - 6 - k = 0\). Next, we combine like terms \(4k - k\) to get \(3k\). This reduction transforms our equation into a much simpler form: \(3k - 6 = 0\). Solving for \(k\):
- Add 6 to both sides: \(3k = 6\)
- Divide both sides by 3: \(k = 2\)
Quadratic Formula
The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a universal method for solving any quadratic equation. In our example, the simplified quadratic equation is: \(2x^2 + 3x - 2 = 0\). Here, \(a = 2\), \(b = 3\), and \(c = -2\). To find the roots of the equation, we substitute these values into the quadratic formula:
First, identify each component:
First, identify each component:
- \(a = 2\)
- \(b = 3\)
- \(c = -2\)
Discriminant
The discriminant \(\Delta\) is part of the quadratic formula that occurs under the square root sign: \(b^2 - 4ac\). It tells us about the nature of the roots of a quadratic equation.
For our equation \(2x^2 + 3x - 2=0\), we calculate the discriminant as follows:
Finally, substitute \(\Delta\) back into the quadratic formula to find the solutions:
For our equation \(2x^2 + 3x - 2=0\), we calculate the discriminant as follows:
- Square the coefficient \(b\): \(3^2 = 9\)
- Multiply \(4\) by \(a\) and \(c\): \(4 \cdot 2 \cdot (-2) = -16\)
- Combine the results: \(9 + 16 = 25\)
Finally, substitute \(\Delta\) back into the quadratic formula to find the solutions:
- \(\frac{-3 \pm \sqrt{25}}{4}\)
- \(\frac{-3 \pm 5}{4}\)
- This results in two roots: \(x = -2\) and \(x = \frac{1}{2}\)
