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The quadratic formula provides a straightforward way to find the solutions, or roots, of a quadratic equation. Quadratic equations are typically in the form \(ax^2 + bx + c = 0\). The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]To use the formula, you first need to identify the coefficients \(a\), \(b\), and \(c\) from the equation.In the equation \(x^2 - 10x + 1 = 0\), we have:

• \(a = 1\)
• \(b = -10\)
• \(c = 1\)

These values are substituted into the quadratic formula to find the roots of the equation. This formula is very powerful because it works for any quadratic equation, regardless of whether it can be factored easily or not.

The discriminant is a key component of the quadratic formula that helps determine the nature of the roots of the quadratic equation. The discriminant is found inside the square root of the quadratic formula and is given by: \[ \Delta = b^2 - 4ac \]Depending on the value of \(\Delta\):

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (a repeated root).
• If \(\Delta < 0\), there are no real roots, but two complex roots.

In our example, the discriminant is calculated as follows:\[ \Delta = (-10)^2 - 4(1)(1) = 100 - 4 = 96 \]Because \(\Delta = 96\) is greater than zero, our quadratic equation \(x^2 - 10x + 1 = 0\) has two distinct real roots.

The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. Once we have calculated the discriminant (\(\Delta\)), we use it to find the exact values of the roots with the quadratic formula. For our equation \(x^2 - 10x + 1 = 0\), we computed that \(\Delta = 96\). Substituting this into the quadratic formula gives us: \[ x = \frac{-(-10) \pm \sqrt{96}}{2(1)} = \frac{10 \pm \sqrt{96}}{2} \] Simplifying further, we get: \[ x = \frac{10 \pm 4\sqrt{6}}{2} = 5 \pm 2\sqrt{6} \] Hence, the roots of the quadratic equation are \(x = 5 + 2\sqrt{6}\) and \(x = 5 - 2\sqrt{6}\). These roots are the points where the quadratic equation crosses the x-axis on a graph. Understanding the roots of quadratic equations is crucial in solving many mathematical problems involving parabolas and other quadratic functions.