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The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. A quadratic equation is typically written in the form of: \[ ax^2 + bx + c = 0 \]. The quadratic formula allows you to solve for the variable \(x\) and is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Here:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

This formula can solve any quadratic equation, where coefficients are numbers. The \(\pm\) symbol indicates there will typically be two solutions.

Solving a quadratic equation usually involves several steps. Let's walk through these steps using the example equation \(2x^2 + 8x + 1 = 0\):1. **Identify the coefficients**: Here, \(a = 2\), \(b = 8\), and \(c = 1\).
2. **Compute the Discriminant**: The discriminant is a key part of the quadratic formula, calculated as \( b^2 - 4ac \). For our example, this gives: \[ 8^2 - 4 \cdot 2 \cdot 1 = 64 - 8 = 56 \]. 3. **Substitute into the Quadratic Formula**: Plugging the values into the quadratic formula we get:\[ x = \frac{-8 \pm \sqrt{56}}{4} \]. 4. **Simplify the Solution**: Break the formula down into simpler parts:\[ x = \frac{-8 \pm 2 \sqrt{14}}{4} \] and eventually: \[ x = \frac{-4 \pm \sqrt{14}}{2} \].5. **Find the Two Solutions**: The two possible solutions are \[ x = -2 + \frac{\sqrt{14}}{2} \] and \[ x = -2 - \frac{\sqrt{14}}{2} \].

The discriminant is a part of the quadratic formula and is very useful in determining the nature of the roots of the quadratic equation. It is given by: \[ b^2 - 4ac \]

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots (not real).

In our example: \[ 8^2 - 4 \cdot 2 \cdot 1 = 56 \] Since the discriminant is positive, we have two distinct real roots.

Sometimes, the solutions to a quadratic equation are irrational numbers, which means they cannot be expressed as a simple fraction. In such cases, we might need to round the solutions.

• **Identify the irrational part**: From our solutions \[ x = -2 + \frac{\sqrt{14}}{2} \] and \[ x = -2 - \frac{\sqrt{14}}{2} \], the irrational part is \(\sqrt{14}\).
• **Approximation**: Calculate the decimal approximation of the irrational number. For \(\sqrt{14} \approx 3.742\).
• **Perform the calculation**: Substitute the approximate value back into the solutions: \[ x \approx -2 + \frac{3.742}{2} \approx -0.129 \] and \[ x \approx -2 - \frac{3.742}{2} \approx -3.871 \].
• **Rounding**: Round these results to the nearest thousandth, which gives the final roots \(x \approx -0.129\) and \(x \approx -3.871\).

Accurate rounding ensures the solutions are practical and easier to interpret.