Problem 33 Find \(b^{2}-4 a c\) and the num... [FREE SOLUTION] (2024)

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When dealing with quadratic equations, one important concept is the discriminant. The discriminant is a specific value calculated from the coefficients of the quadratic equation and is critical in determining the nature of the equation's solutions. For any quadratic equation of the formewline ewline ewline ewline \[ax^2 + bx + c = 0\],ewline ewline the discriminant is given by the formula:ewline ewline \[b^2 - 4ac\].ewline ewline nTo find the discriminant, follow these steps:ewline

• Identify the coefficients 'a', 'b', and 'c'.
• Substitute these values into the formula \[b^2 - 4ac\].
• Simplify the expression to get the discriminant.

In the equation \[x^2 - 6x + 2 = 0\], we identified the coefficients as \(a = 1\), \(b = -6\), and \(c = 2\). Substituting those values into the formula, we get:\[(-6)^2 - 4(1)(2) = 36 - 8 = 28\].This value of 28 is the discriminant.

The discriminant not only helps in solving a quadratic equation but also provides insight into the nature of its roots. By analyzing the discriminant value, we can directly determine the number of real solutions for the quadratic equation. Here’s how:

• If the discriminant \(b^2 - 4ac > 0\), the equation has 2 distinct real solutions.
• If the discriminant \(b^2 - 4ac = 0\), the equation has 1 real solution (also called a repeated or double root).
• If the discriminant \(b^2 - 4ac < 0\), the equation has no real solutions. Instead, it has 2 complex solutions.

In the example equation \[x^2 - 6x + 2 = 0\], we found the discriminant to be 28, which is greater than 0. Therefore, the equation \(x^2 - 6x + 2 = 0\) has 2 distinct real solutions. This tells us that the graph of the quadratic function \(y = x^2 - 6x + 2\) will intersect the x-axis at two points.

The quadratic formula is a robust tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is derived from the process of completing the square and can be stated as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where:

• \(x\) represents the solutions to the quadratic equation,
• \(a\), \(b\), and \(c\) are the coefficients from the equation,
• \(\pm\) indicates that there will often be two solutions: one involving addition and the other subtraction.

Using our example \(x^2 - 6x + 2 = 0\), we substitute \(a = 1\), \(b = -6\), and \(c = 2\) into the quadratic formula:\[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(2)}}{2(1)}\],which simplifies to:\[x = \frac{6 \pm \sqrt{28}}{2}\],and then further simplifies to two solutions:\[x = \frac{6 + \sqrt{28}}{2}\] \text{ and } \[x = \frac{6 - \sqrt{28}}{2}\].This process confirms that there are indeed two distinct real solutions for the quadratic equation, as we expected from our discriminant calculation.