91Ó°ÊÓ

Understanding how to solve quadratic equations is fundamental to algebra. These are equations of the second degree, meaning they have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). A widely used method to solve for \(x\) is applying the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The solution involves three primary steps. First, identify the coefficients from the quadratic equation. Next, plug these values into the quadratic formula. Lastly, calculate the value of \(x\) by evaluating the formula, including simplifying the discriminant, which involves square rooting the expression \(b^2 - 4ac\).

For the equation \(3x^2 + x - 1 = 0\), applying these steps will yield two possible solutions for \(x\), which comes from the 'plus-minus' part of the quadratic formula. Students should practice these steps to become proficient in solving quadratic equations across various examples.

The discriminant is a valuable component in quadratic equations and plays a central role in determining the nature and number of solutions a quadratic equation will have. It is represented as \(b^2 - 4ac\) in the quadratic formula. The discriminant can tell us if the solutions are real or complex and whether they are unique or repeated.

For instance, if the discriminant is positive, there are two distinct real solutions. If it equals zero, there is exactly one real solution (repeated roots). If the discriminant is negative, the solutions are complex, which means they involve imaginary numbers. When working on \(3x^2 + x - 1 = 0\), we find the discriminant to be \(1 + 12\) or \(13\), which is positive; hence, there are two distinct real solutions.

The coefficients in a quadratic equation—commonly denoted as \(a\), \(b\), and \(c\)—are crucial in shaping the curve of the quadratic graph and determining the nature of its roots. \(a\), the coefficient of \(x^2\), influences the direction of the parabola (whether it opens upwards or downwards) and the width of the curve. Coefficient \(b\) impacts the position of the vertex along the horizontal axis, while \(c\), the constant term, determines the y-intercept where the parabola crosses the y-axis.

In the equation \(3x^2 + x - 1 = 0\), \(a = 3\), \(b = 1\), and \(c = -1\). The positive value of \(a\) indicates that the parabola opens upwards, and since \(a > 1\), the parabola is narrower compared to the standard parabola \(y = x^2\). Understanding how these coefficients affect the equation can assist in graphing the quadratic and predicting the nature of its roots even before applying the quadratic formula.