Question

\(g(x)=-3 x^2-6 x+5\)

Short answer

The roots of the equation \(g(x) = -3 x^2 - 6 x + 5\) are \(x = -5\) and \(x = 3\).

Step-by-step solution

Identify a, b and c

Identify the coefficients \(a\), \(b\), and \(c\) from the given equation. Here, \(a = -3\), \(b = -6\), and \(c = 5\).

Substitute in the Quadratic Formula

Substitute the coefficients into the quadratic formula: \(x = [-(-6) \pm \sqrt{(-6)^2 - 4*(-3)*5}] / (2*-3)\).

Simplify the Expression

Simplify the expression under the square root (the discriminant) and the rest of the formula to get \(x = [6 \pm \sqrt{36+60}]/-6 = [6 \pm \sqrt{96}]/-6\).

Solve for x

Solve for the two possible values of \(x\) to get \(x = [6 + \sqrt{96}]/-6 = -1 - 4 = -5\) and \(x = [6 - \sqrt{96}]/-6 = -1 + 4 = 3\)

Key concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations, which are polynomial equations of the second degree. These take the general form: \(ax^2 + bx + c = 0\). When we talk about solving these equations, we are looking for the values of \(x\) that make the equation true. The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

You can think of the formula as a recipe that helps you find the "roots" or solutions directly, with the only ingredients being the coefficients \(a\), \(b\), and \(c\) from your quadratic equation. Knowing this formula enables you to find solutions even when factoring is not possible or is too complex.

Discriminant

The Discriminant is the part that sits inside the square root of the quadratic formula: \(b^2 - 4ac\). It's crucial because it tells us how many real roots a quadratic equation has.

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a repeated root).
• If \(b^2 - 4ac < 0\), there are no real roots, only complex ones.

In the exercise, the discriminant calculation was \(36 + 60 = 96\), which is greater than zero. Therefore, the quadratic equation in question has two distinct real roots. Understanding the discriminant can save time by allowing you to know how many solutions you should expect, right from the start.

Polynomial Coefficients

In a quadratic equation, the polynomial coefficients \(a\), \(b\), and \(c\) play vital roles. These coefficients are the numbers that multiply the terms of the polynomial equation \(ax^2 + bx + c = 0\). Identifying these coefficients is often the first step in solving a quadratic equation using the quadratic formula.

• \(a\) is the coefficient of \(x^2\).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term, or the term with no \(x\) attached to it.

In the provided exercise, we had \(a = -3\), \(b = -6\), and \(c = 5\). These values are then plugged into the quadratic formula to proceed with solving the equation. Mistakenly identifying these coefficients can lead to an incorrect solution, so it's crucial to get the right values.

Roots of a Polynomial

The roots of a polynomial are the solutions of the equation where the polynomial is equal to zero, meaning they are the values of \(x\) that satisfy \(ax^2 + bx + c = 0\). For quadratic equations, we typically expect up to two roots, found using the quadratic formula.In the exercise, the roots of the polynomial \(g(x) = -3x^2 - 6x + 5\) were determined as \(x = -5\) and \(x = 3\). These roots are calculated by evaluating the quadratic formula with the given coefficients. Knowing the roots is essential because they reveal the values which make the polynomial null, offering critical insights into the behavior of the quadratic function. They represent points where the graph of the quadratic equation crosses the x-axis. Having a good grasp of finding and interpreting roots enriches your understanding of quadratic equations immensely.