91Ó°ÊÓ

Understanding how to solve quadratic equations is a fundamental skill in algebra. A quadratic equation is any equation that can be rewritten in the standard form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are coefficients and \(a\) is not equal to zero. To solve such an equation, one can use the quadratic formula, which provides a solution regardless of whether the quadratic is factorable. The formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). The \(\pm\) symbol indicates that there are two possible solutions for \(x\), known as the roots of the equation. These roots can be real or complex numbers, depending on the discriminant value.
The steps to solve the quadratic equation we encounter, which is \(x^2 + 5x + 3 = 0\), are systematic. We identify the coefficients \(a = 1\), \(b = 5\), and \(c = 3\) and then substitute them into the quadratic formula. The process will lead us to the two possible values for \(x\) that satisfy the equation.

The discriminant is a powerful tool in determining the nature of the roots without actually solving the equation. It is represented by the symbol \(d\) and is found using the formula \(d = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the numeric coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
For our specific problem, we calculate the discriminant by plugging the coefficients into the formula: \(d = 5^2 - 4(1)(3) = 25 - 12 = 13\). The discriminant informs us about the roots in the following ways: if \(d > 0\), there are two distinct real roots; if \(d = 0\), there is exactly one real root (a repeated root); if \(d < 0\), there are no real roots, but rather two complex roots. Since our discriminant is 13, a positive number, we know our equation has two distinct real roots.

The roots of a quadratic equation are the solutions to the equation. When we talk about roots, we are referring to the values of \(x\) that make the equation true, or simply put, where the equation equals zero. These are the values we find using the quadratic formula. In the context of our problem, \(x^2 + 5x + 3 = 0\), we use the calculated discriminant \(d = 13\) to find the roots.
By substituting the discriminant and the coefficients into the quadratic formula, we get two roots: \(x_1 = \frac{{-b + \sqrt{d}}}{{2a}}\) and \(x_2 = \frac{{-b - \sqrt{d}}}{{2a}}\), which simplifies to \(x_1 = \frac{{-5 + \sqrt{13}}}{{2}}\) and \(x_2 = \frac{{-5 - \sqrt{13}}}{{2}}\). These roots are the values of \(x\) that satisfy the quadratic equation. It's essential to understand both how to compute these roots and the meaning behind them to grasp thoroughly the concepts of quadratic equations.