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Chapter 0: Problem 82

Solve each quadratic equation using the quadratic formula. $$3 x^{2}=6 x-1$$

Short Answer

Expert verified
The solutions to the equation are \( x_{1} = 1 + \sqrt{2} \) and \( x_{2} = 1 - \sqrt{2} \)

Step by step solution

01

Convert to Quadratic Standard Form

First, the given quadratic equation, \( 3x^{2} = 6x - 1 \), must be arranged in the standard form of \( ax^{2} + bx + c = 0 \). By subtracting \( 6x \) and adding \( 1 \) both sides of the equation, it becomes \( 3x^{2} - 6x + 1 = 0 \)
02

Identify the Coefficients

Once the equation is in quadratic standard form, the coefficients for \( a \), \( b \), and \( c \) can be identified. For the equation \( 3x^{2} - 6x + 1 = 0 \), the coefficients are \( a = 3 \), \( b = -6 \), and \( c = 1 \)
03

Apply the Quadratic Formula

Use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \), and the values of \( a \), \( b \), and \( c \) to find the roots of the quadratic equation. Plugging in the identified values, the solution will be \( x = \frac{-(-6) \pm \sqrt{(-6)^{2}-4(3)(1)}}{2*3} \) = \( \frac{6 \pm \sqrt{36-12}}{6} \)
04

Simplify the Quadratic Formula

Finally, simplify the solution within the square root and the fraction to get the final answer. The square root part simplifies to \( \sqrt{24} \), so the entire thing simplifies to \( x = \frac{6 \pm \sqrt{24}}{6} \). Now the equation can be splitted into two possibilities by the \( \pm \) sign: When \( + \sqrt{24} \): \( x_{1} = \frac{6 + \sqrt{24}}{6} = 1+ \sqrt{2} \), and, when \( - \sqrt{24} \): \( x_{2} = \frac{6 - \sqrt{24}}{6} = 1- \sqrt{2} \)

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