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The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is an equation which can be written in the form \(ax^2 + bx + c = 0\). The standard form is necessary to apply the quadratic formula. The formula itself is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The term inside the square root, \( b^2 - 4ac \), is known as the discriminant. The symbols \( \pm \) mean there are generally two solutions: one for the plus sign and one for the minus sign.
This formula gives you all possible solutions for the quadratic equation by taking the coefficients \(a\), \(b\), and \(c\) from the standard form.
The formula covers all cases, whether the solutions are real or complex.

The discriminant, represented as \( \Delta \), is a key part of the quadratic formula. It is found inside the square root:\[ \Delta = b^2 - 4ac \]
Depending on its value, the discriminant reveals important information about the solutions of the quadratic equation:

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution (a repeated solution).
• If \( \Delta < 0 \), there are two complex solutions.

In the example equation, \(16x^2 + 8x + 1 = 0\), the discriminant is \(0\). This tells us there is exactly one real solution.

Before solving a quadratic equation using the quadratic formula, it must be written in the standard form:\[ax^2 + bx + c = 0\]
Here, \(a\), \(b\), and \(c\) are constants. The expression on the left-hand side should be zero on its own. Identifying \(a\), \(b\), and \(c\) correctly is essential for applying the quadratic formula.
In our case, the equation \(16x^2 + 8x + 1 = 0\) is already in standard form with \(a = 16\), \(b = 8\), and \(c = 1\). These values are directly plugged into the quadratic formula.