Solve each equation using completing the square. See Examples 3 and 4. 3x^2 - 9x + 7 = 0

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1. \[ 3x^2 - 9x = -7 \]
2. \[ 3(x^2 - 3x) = -7 \]
3. \[ 3(x^2 - 3x + \left(\frac{-3}{2}\right)^2) = -7 + 3\left(\frac{-3}{2}\right)^2 \]
4. \[ 3(x - \frac{3}{2})^2 = -7 + \frac{27}{4} \]

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves rearranging the equation and adding a specific value to both sides to create a square of a binomial. This technique simplifies the process of finding the roots of the equation, making it easier to solve for the variable.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, using the quadratic formula, or completing the square. Understanding the standard form of a quadratic equation is essential for applying these methods effectively.

Discriminant

The discriminant is a component of the quadratic formula, given by the expression b^2 - 4ac. It determines the nature of the roots of a quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. Analyzing the discriminant helps in understanding the solutions' characteristics before solving the equation.