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The quadratic formula is a powerful tool for solving quadratic equations. It provides a way to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The formula is expressed as:
\[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]
Using the plus-minus (\( \pm \)) symbol, it gives two possible solutions for the value of \( x \). This formula works for all quadratic equations, making it an essential part of algebra.

To solve a quadratic equation using the quadratic formula, first identify the coefficients \( a \), \( b \), and \( c \). Then, substitute these values into the formula and simplify.

Before applying the quadratic formula, it's important to write the equation in its standard form:
\[ ax^2 + bx + c = 0 \]
This format ensures that the quadratic formula can be used correctly. Here, the coefficients \( a \), \( b \), and \( c \) must be correctly identified. For example, in the equation \( 2x^2 - 2x = 1 \), we first move all terms to one side to obtain:
\[ 2x^2 - 2x - 1 = 0 \]
Now the equation is in the correct standard form, ready for identifying the coefficients and applying the quadratic formula.

For a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the coefficients \( a \), \( b \), and \( c \) are crucial for solving the equation. They represent the numerical values that multiply the variables \( x^2 \), \( x \), and the constant term, respectively. In the equation \( 2x^2 - 2x - 1 = 0 \):

• \( a = 2 \)
• \( b = -2 \)
• \( c = -1 \)

These coefficients are substituted into the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions for \( x \). Ensure you always identify these coefficients correctly to avoid errors in solving the equation.

Once the coefficients are substituted into the quadratic formula, the next step is to simplify the expression to find the solutions. For instance, substituting \( a = 2 \), \( b = -2 \), and \( c = -1 \) into the quadratic formula gives:
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(-1)}}{2(2)} \]

Simplify inside the square root first:
\[ x = \frac{2 \pm \sqrt{4 + 8}}{4} \]

Then, simplify the square root and the fraction:
\[ x = \frac{2 \pm \sqrt{12}}{4} \]
\[ x = \frac{2 \pm 2\sqrt{3}}{4} \]
\[ x = \frac{1 \pm \sqrt{3}}{2} \]

Thus, the solutions are:
\[ x = \frac{1 + \sqrt{3}}{2} \]
and
\[ x = \frac{1 - \sqrt{3}}{2} \]
Simplifying correctly is crucial for finding the accurate roots of the equation.