Question

\(g(x)=3 x^2+18 x-5\)

Short answer

The roots of the quadratic equation \(g(x)=3 x^2+18 x-5\) are approximately -0.74 and 2.23.

Step-by-step solution

Identify the coefficients

The coefficients of the equation are \(a = 3\), \(b = 18\), and \(c = -5\).

Compute the discriminant

The discriminant is \(D = b^2-4ac = (18)^2 - 4*3*(-5) = 324+60 = 384\).

Apply Quadratic formula

The roots of the equation can be found using the quadratic formula, \(x=\frac{-b \pm \sqrt{D}}{2a}\). Substituting the given values, we get \(x1=\frac{-18+ \sqrt{384}}{2*3}\) and \(x2=\frac{-18- \sqrt{384}}{2*3}\). Calculation gives the roots \(x1 \approx -0.74 \) and \(x2 \approx 2.23 \)

Key concepts

Discriminant

The discriminant is a fundamental concept in understanding quadratic equations. It helps determine the nature of the roots of a quadratic equation, given by the formula:\[D = b^2 - 4ac\]where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation in the form \(ax^2 + bx + c = 0\). The discriminant tells us:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, often called a repeated or double root.
• If \(D < 0\), there are no real roots, but two complex roots.

For the equation \(g(x)=3x^2+18x-5\), the discriminant was calculated as \(384\). Because \(384 > 0\), it confirms the presence of two real roots for this equation.

Quadratic Formula

Once you know the discriminant, the quadratic formula is a powerful tool to find the actual roots of the quadratic equation. The quadratic formula is expressed as:\[x = \frac{-b \pm \sqrt{D}}{2a}\]Here, \(b\) is the coefficient of \(x\), \(a\) is the coefficient of \(x^2\), and \(D\) is the discriminant.
The "\(\pm\)" sign indicates that there are typically two solutions: one with a plus and one with a minus. This feature allows us to compute both roots of the equation, making it extremely useful for solving any quadratic equation.
In the context of our original problem, substituting \(b = 18\), \(a = 3\), and \(D = 384\) into the quadratic formula, we derive the calculations leading to the roots of the equation:

Roots of Equations

Finding the roots of a quadratic equation means determining the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots represent the points where the graph of the equation intersects the x-axis.
Using the quadratic formula, our previous example produced two roots, \(x_1\) and \(x_2\). By plugging in the known values into the quadratic formula:\[x_1 = \frac{-18 + \sqrt{384}}{2*3}\]\[x_2 = \frac{-18 - \sqrt{384}}{2*3}\]After performing the calculations:

• \(x_1 \approx -0.74\)
• \(x_2 \approx 2.23\)

This confirms that the quadratic equation \(g(x) = 3x^2 + 18x - 5\) intersects the x-axis at these points, providing two distinct real roots.