Solve each equation by the method of your choice. $3 x^{2} - 7 x + 1 = 0$

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Identify the given quadratic equation: $3x^2 - 7x + 1 = 0$.

Recall that a quadratic equation in the form $ax^2 + bx + c = 0$ can be solved using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Substitute the coefficients from the equation into the quadratic formula: $a = 3$, $b = -7$, and $c = 1$, so the formula becomes $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$.

Simplify inside the square root (the discriminant): calculate $b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot 1$.

Evaluate the expression under the square root and then compute the two possible values for $x$ by applying the plus and minus signs in the quadratic formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and typically has two solutions, which can be real or complex numbers.

Factoring and the Zero Product Property

Factoring involves rewriting a quadratic equation as a product of two binomials. The Zero Product Property states that if the product of two factors is zero, then at least one factor must be zero, allowing us to solve for the variable.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides a method to find the roots of any quadratic equation. It is especially useful when factoring is difficult or impossible, and the discriminant (b² - 4ac) determines the nature of the roots.

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Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

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