Problem 64 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$2 x^{2}-2 x-12=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(2x^2 - 2x -12 = 0\) are \(x = 3\) and \(x = -2\).

Step 1: Identify a, b, c

From the given equation, one can identify the coefficients as \(a = 2\), \(b = -2\), and \(c = -12\)

Step 2: Substitute values into the quadratic equation

Substitute the values of a, b, c into the quadratic formula. Thus, \(x=\frac{-(-2)\pm\sqrt{(-2)^{2}-4*2*(-12)}}{2*2}\)

Step 3: Simplify the equation

Simplify the formula will give \(x = \frac{2\pm\sqrt{4+96}}{4}\)

Step 4: Find the roots

The solutions step by step will be \(x = \frac{2\pm\sqrt{100}}{4} ----> x=\frac{2\pm10}{4}\). Therefore, the roots of the equation are \(3\) and \(-2\).

Key Concepts

Quadratic Equation

A quadratic equation is an equation of degree 2, which means the highest power of the variable (usually denoted as \(x\)) is two. It is usually in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This structure is what differentiates quadratic equations from linear equations (which are of degree 1).
To solve a quadratic equation, we aim to find the values of \(x\) that satisfy the equation, i.e., make the equation equal to zero. These values of \(x\) are known as the roots of the equation.

• Main Parts: The equation consists of three main parts: the quadratic term \(ax^2\), the linear term \(bx\), and the constant term \(c\).
• Purpose: We often deal with quadratic equations in problems involving area, projectile motion, and the relationship between variables.
• Properties: Quadratic equations can have either two distinct real roots, one real root (also called a repeated or double root), or two complex roots.

Roots of an Equation

The roots of a quadratic equation are the solutions to the equation, the values of \(x\) that satisfy the equation. Finding these roots helps us understand important features of the quadratic function.
Most quadratic equations can be solved using the quadratic formula, which is derived from completing the square and looks like this:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• Two Real Roots: If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• One Real Root: If \(b^2 - 4ac = 0\), there is one repeated real root.
• Complex Roots: If \(b^2 - 4ac < 0\), the roots are complex.

Using the quadratic formula not only gives us the roots but also hints at the nature of the roots through the term under the square root, known as the discriminant \(b^2 - 4ac\). A positive discriminant indicates two distinct real solutions, zero means a single solution, and a negative means complex solutions.

Coefficient Identification

Before solving a quadratic equation using the quadratic formula, it is crucial to correctly identify the coefficients \(a\), \(b\), and \(c\). These coefficients are simply the numbers in front of each term in the standard form of a quadratic equation \(ax^2 + bx + c = 0\).
In the equation \(2x^2 - 2x - 12 = 0\), identifying these coefficients accurately ensures a successful solution.

• Quadratic Coefficient: Here, \(a = 2\). The coefficient \(a\) affects the "width" and direction of the parabola formed by the quadratic equation. If \(a\) is negative, the parabola opens downward; if positive, upward.
• Linear Coefficient: The value of \(b = -2\) serves as the linear component of the equation. It affects the symmetry of the parabola concerning its vertical axis.
• Constant Term: The constant \(c = -12\) provides the vertical translation (shift up or down) of the parabola.

Correctly identifying these components allows for correct insertion into the quadratic formula to solve the equation properly.