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The Quadratic Formula is a powerful tool for solving quadratic equations. A quadratic equation has the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents the variable. The Quadratic Formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps you find the solutions, or roots, of any quadratic equation. It's particularly useful when the equation cannot be easily factored. The term under the square root, \( b^2 - 4ac \), is called the discriminant, and it determines how many solutions the equation has:

• If the discriminant is positive, you get two distinct real roots.
• If it is zero, there is exactly one real root (which occurs in our original exercise).
• If it is negative, the equation has two complex roots.

Understanding the Quadratic Formula is crucial, as it allows you to solve quadratics even in complicated situations.

Solving quadratic equations involves finding the value(s) of \( x \) that satisfy the equation. Using the Quadratic Formula is just one of many methods to solve them. Here are some common methods:

• Factoring involves expressing the quadratic as a product of two binomial expressions, making it easier to solve.
• Completing the Square transforms the quadratic into a perfect square trinomial, which can be solved by taking square roots.
• Graphing lets you visualize the solutions as the points where the graph of \( y = ax^2 + bx + c \) intersects the x-axis.

Choosing the right method depends on the specific equation. Sometimes, certain methods are more straightforward, while others may require more steps, like completing the square, which can be a bit meticulous. It's essential to understand each method well so you can select the most efficient one for the problem you are solving.

Factoring is one of the most intuitive and preferred methods for solving simple quadratic equations. When you factor a quadratic equation like \( ax^2 + bx + c = 0 \), you're expressing it as \((mx + n)(px + q) = 0\). This method relies on finding two numbers that multiply to \( ac \) and add to \( b \). Although it may not be applicable to every quadratic equation, here’s a quick guide:

• Start by multiplying \( a \) and \( c \).
• Find two numbers that multiply to \( ac \) and add up to \( b \).
• Rewrite the middle term, \( bx \), using the two numbers found.
• Factor by grouping, splitting the quadratic into two binomials.

For the equation in the original exercise, \( x^2 - 14x + 49 = 0 \), it's actually a perfect square trinomial where \((x-7)^2 = 0\). This shows that both roots are equal, \( x = 7 \), confirming the solution found using the Quadratic Formula. Familiarizing yourself with factoring will make solving such equations more efficient and insightful.