Problem 69 - Algebra and Trigonometry

Question

Solve equation using the quadratic formula. $$ 3 x^{2}-3 x-4=0 $$

Step-by-Step Solution

Answer

The solutions for the given quadratic equation are \(x = \frac{3 + \sqrt{57}}{6}\) and \(x = \frac{3 - \sqrt{57}}{6}\).

Step 1: Identify the values of a, b, and c

From the given equation \(3x^{2} - 3x - 4 = 0\), the values of the coefficients are \(a = 3\), \(b = -3\) and \(c = -4\).

Step 2: Substitute into the quadratic formula

Substitute the values into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). This results in \(x = \frac{-(-3) \pm \sqrt{(-3)^{2}-4*3*(-4)}}{2*3}\).

Step 3: Simplify the equation

Now, perform computations. This will eventually give the two solutions for the equation: \(x = \frac{3 \pm \sqrt{9+48}}{6}\).\nFurther simplifying gives \(x = \frac{3 \pm \sqrt{57}}{6}\). The two solutions are thus \(x = \frac{3 + \sqrt{57}}{6}\) and \(x = \frac{3 - \sqrt{57}}{6}\).

Key Concepts

Solving Quadratic Equations

Understanding quadratic equations is a fundamental skill in algebra. A quadratic equation is typically written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(x\) represents the unknown variable. The most reliable method for solving any quadratic equation is using the quadratic formula, which provides a straightforward solution for finding the values of \(x\) that satisfy the equation.

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula derives from completing the square in the general quadratic equation and expresses the solution in terms of the coefficients \(a\), \(b\), and \(c\). To solve a specific quadratic equation, you need to identify these coefficients and substitute them into the formula. The \(\pm\) symbol indicates that there are generally two solutions to a quadratic equation, which account for the parabolic shape of their graphs. Upon simplifying, you'll find the possible values of \(x\) that make the original equation true.

Quadratic Equation Roots

The roots of a quadratic equation refer to the values of \(x\) that make the equation equal to zero - these are also known as the solutions of the equation. They are the points where the graph of the equation, a parabola, intersects the \(x\)-axis. Using the quadratic formula, we can systematically find these roots without visualization.

The term under the square root in the quadratic formula, \(b^2 - 4ac\), is known as the discriminant. It is a key player in determining the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it is zero, there is one real root (or two identical real roots). If the discriminant is negative, the roots are complex. In the case of the provided exercise, the discriminant is positive (\(57\)), indicating two distinct real roots, \(\frac{3 + \sqrt{57}}{6}\) and \(\frac{3 - \sqrt{57}}{6}\).

Coefficient Identification in Quadratics

Correctly identifying the coefficients in a quadratic equation is crucial to apply the quadratic formula effectively. Coefficients are the numerical factors that multiply the variable terms. In the standard form of a quadratic equation, \(ax^2 + bx + c = 0\), \(a\) is the coefficient of the \(x^2\) term, \(b\) is the coefficient of the \(x\) term, and \(c\) is the constant term.

In our example, the equation \(3x^2 - 3x - 4 = 0\) has \(a = 3\), \(b = -3\), and \(c = -4\). Recognizing the signs of these coefficients is as important as recognizing their magnitude. It is the values of these coefficients that we plug into the quadratic formula to find the roots of the equation. Any error in identifying these coefficients may lead to incorrect solutions, so always double-check your identification before proceeding with the quadratic formula.