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Chapter 11: Problem 4

Find the error in each, and correct the mistake. The discriminant of \(3 z^{2}-4 z+1=0\) is $$ \begin{aligned} \sqrt{b^{2}-4 a c} &=\sqrt{(-4)^{2}-4(3)(1)} \\ &=\sqrt{16-12} \\ &=\sqrt{4} \\ &=2 \end{aligned} $$

Short Answer

Expert verified
The error in the given solution is that the square root is taken for the discriminant, which is incorrect. The correct discriminant for the quadratic equation \(3z^2 - 4z + 1 = 0\) is \(\Delta = 4\).

Step by step solution

01

Identify the coefficients of the quadratic equation

In the given equation, \(3z^2 - 4z + 1 = 0\), the coefficients are: \(a = 3\), \(b = -4\), and \(c = 1\).
02

Calculate the discriminant

The discriminant, \(\Delta\), is given by the formula \(\Delta = b^2 - 4ac\). Using the identified coefficients, we can calculate the correct discriminant: \[ \begin{aligned} \Delta = b^2 - 4ac &= (-4)^2 - 4(3)(1) \\ &= 16 - 12 \\ &= 4 \end{aligned} \]
03

Present the corrected solution

The correct discriminant for the quadratic equation \(3z^2 - 4z + 1 = 0\) is \(\Delta = 4\), not \(\sqrt{4} = 2\) as shown in the given solution.

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SOLVE. \(4=9 r^{2}\)
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