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The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps solve for \(x\), the unknown variable in the equation. Here's how the formula works:

• The "\(-b \pm\)" part means that there are typically two solutions: one adds \(\sqrt{b^2 - 4ac}\) and the other subtracts it.
• \(b^2 - 4ac\) is a critical component known as the "discriminant," determining the nature of the roots.
• Divide by \(2a\) ensures that the solutions are correctly scaled according to the equation's leading coefficient \(a\).

Using the quadratic formula saves time and avoids lengthy algebra, especially in complex equations. It reveals essential mathematical properties about the equation's solutions through the discriminant, making it indispensable for solving quadratic equations efficiently.

The discriminant is a part of the quadratic formula found under the square root symbol \(b^2 - 4ac\). Its value tells us how many and what type of roots we can expect from the quadratic equation \(ax^2 + bx + c = 0\).

• If \(b^2 - 4ac > 0\), there are two distinct real roots. This occurs when the equation crosses the x-axis at two different points.
• If \(b^2 - 4ac = 0\), there is exactly one real root, also known as a repeated or double root. This means the parabola touches the x-axis at one point.
• If \(b^2 - 4ac < 0\), there are two complex roots, meaning the equation does not intersect the x-axis, and the solutions involve imaginary numbers.

In the given problem, the discriminant is computed as \(441\), which is greater than zero. This tells us that the quadratic equation has two distinct real roots. Knowing the discriminant's value provides crucial insight into the nature of the solutions before we even compute them.

The "roots" of a quadratic equation are the values of \(x\) that satisfy \(ax^2 + bx + c = 0\). They are the solutions that make the equation true by reducing it to zero. In graphical terms, these roots correspond to the x-coordinates where the parabola representing the equation intersects the x-axis.To find the roots using the quadratic formula, substitute the values of the coefficients \(a\), \(b\), and \(c\) into the formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In the problem you're tackling, the values are:

• \(a = 10\)
• \(b = -29\)
• \(c = 10\)

By calculating, the roots come out to be \(x = 2.5\) and \(x = 0.4\). These roots can be verified by plugging them back into the original equation. For instance:

• When \(x = 2.5\), substitute into the equation and check if it equals zero.
• When \(x = 0.4\), do the same to ensure accuracy.

These calculations verify that both solutions satisfy the original equation, confirming the correctness of the given solutions.