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To solve a quadratic equation effectively, it's crucial to first convert it into the standard form. The standard form of a quadratic equation is:\[ ax^2 + bx + c = 0 \]Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. This format is essential because it allows you to systematically apply solving techniques like factoring, completing the square, or using the quadratic formula.
In the given equation \(5x^2 = 6 - 13x\), we needed to rearrange it by moving all terms to one side. This conversion gave us \(5x^2 + 13x - 6 = 0\), making it clear that \(a = 5\), \(b = 13\), and \(c = -6\).
Having a quadratic in standard form simplifies solving and analyzing the equation, which is important for finding the solution efficiently.

The quadratic formula is a powerful tool for finding the roots of any quadratic equation in standard form. It is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula arises from the method of completing the square and provides a universal solution pathway.
You simply substitute the coefficients \(a\), \(b\), and \(c\) derived from your equation into the formula. This substitution allows you to solve even complex quadratic equations without much hassle.
In our example equation, plugging in \(a = 5\), \(b = 13\), and \(c = -6\) into the formula, we performed: \[ x = \frac{-13 \pm \sqrt{13^2 - 4 \times 5 \times (-6)}}{2 \times 5} \].
The quadratic formula rescues us especially when other methods like factoring aren't straightforward, ensuring solutions are found for any quadratic equation.

To solve a quadratic equation, following a step-by-step approach is vital. Here's a simple guide on how to tackle it:

• First, ensure the quadratic is in standard form: \(ax^2 + bx + c = 0\).


• Identify the coefficients \(a\), \(b\), and \(c\) from your equation.


• Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].


• Substitute the values of \(a\), \(b\), and \(c\) into the formula.


• Calculate the discriminant \(b^2 - 4ac\). If it's positive, you'll have two real solutions; if zero, one solution; if negative, the solutions are complex numbers.


• Simplify the expression to find the values of \(x\).

In the given problem, after substituting into the quadratic formula and simplifying, we found two solutions: \(x = -1\) and \(x = 1.2\).
This methodical pattern ensures clarity and precision in solving even the most challenging quadratic problems.