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The quadratic formula is a powerful tool used for finding the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). This is essential because it provides a solution even when the roots are not easily factorable. The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's how it works:

• \(b\) is the coefficient of \(x\), \(a\) is the coefficient of \(x^2\), and \(c\) is the constant term.
• The \(\pm\) sign indicates there are potentially two solutions, one involving addition and the other involving subtraction.
• "\(\sqrt{b^2 - 4ac}\)" is called the discriminant, which we'll discuss more below.

Using this formula makes it possible to solve for \(x\) no matter the values of \(a\), \(b\), and \(c\). In our problem, by identifying \(a = -2\), \(b = 7\), and \(c = 4\), you can substitute these into the formula, making it straightforward to calculate the roots.

The discriminant is a critical part of the quadratic formula, represented by \(b^2 - 4ac\). It helps determine the nature of the roots of the quadratic equation without actually solving it. Here's what the discriminant tells us:

• If \(\Delta > 0\), the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two points.
• If \(\Delta = 0\), there is exactly one real root, often referred to as a repeated or double root. The parabola just touches the x-axis at one point (the vertex).
• If \(\Delta < 0\), there are no real roots. Instead, the roots are complex numbers, indicating that the parabola does not intersect the x-axis.

For our specific equation, the discriminant \(\Delta = 49 + 32 = 81\) is positive, so we expect two distinct real roots. This allows us to apply the quadratic formula with these roots as solutions.

Real roots refer to the solutions of a quadratic equation that are real numbers. In the context of quadratic functions, these real roots represent the x-values where the parabola crosses the x-axis. Using the discriminant and the quadratic formula, we can mathematically predict and calculate these intersections.
In our example, since the discriminant is positive, we substitute \(\Delta = 81\) back into the quadratic formula:\[ x = \frac{-7 \pm \sqrt{81}}{-4} \] This becomes two separate calculations:

• For the first root, use the positive square root: \(x_1 = \frac{-7 + 9}{-4} = -\frac{1}{2}\)
• For the second root, use the negative square root: \(x_2 = \frac{-7 - 9}{-4} = 4\)

Thus, the real roots of the function \(f(x) = -2x^2 + 7x + 4\) are \(x = -\frac{1}{2}\) and \(x = 4\). Finding these roots helps us understand the behavior of the quadratic function, specifically how it interacts with the x-axis.