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A quadratic equation is a polynomial equation of the second degree. It generally takes the form: \[ ax^2 + bx + c = 0 \]where * **a**, **b**, and **c** are constants * **x** is the variableIn the provided exercise, the quadratic equation we are solving is: \[ 6x^2 = 36 \]A quadratic equation can have either two solutions, one solution, or no real solutions, depending on the discriminant (\[ b^2 - 4ac \]). For our exercise, it simplifies to solving \[ x^2 = 6 \].Understanding this concept will help you apply similar methods to other equations. Always remember the goal is to isolate the variable and solve for **x**.

Square roots are essential in solving quadratic equations where the variable is squared. The square root of a number **a** is a value that, when multiplied by itself, gives **a**. It is represented as: \[ \text{If } y = \text{√a, then } y^2 = a \]In solving \[ 6x^2 = 36 \], we isolate **x^2** to get \[ x^2 = 6 \], and then proceed to take the square root of both sides. This step is crucial because:* It allows us to find the values of **x** that satisfy the equation. * Remember, each square root has both a positive and a negative solution, so we write: \[ x = \text{±√6} \]Taking the square root can transform a quadratic equation into a simpler form and lead us directly to the solution.

Following proper steps is important for solving equations systematically. Here are the steps to solve \[ 6x^2 = 36 \]:1. **Divide to Isolate the Variable**: First, divide both sides by 6. This simplifies our equation to \[ x^2 = 6 \]. By dividing, we make the coefficient of \[ x^2 \] equal to 1 which simplifies our next steps. 2. **Take the Square Root**: Next, take the square root of both sides to get \[ x = \text{±√6} \]. This gives us the final solutions because the square root function undoes squaring. Following these steps ensures that we systematically reduce the original equation to a simple form that we can solve. Always ensure each step is clear and justified to prevent errors.