Problem 154 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation. Round your answer to three decimal places, if necessary. $$2 x^{2}+x-4=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0.782\) and \(x = -2.282\).

Step 1: Identify the coefficients

In the equation \(2x^2 + x - 4 = 0\), the coefficients are \(a = 2\), \(b = 1\), and \(c = -4\).

Step 2: Apply the quadratic formula

Plug these values into the quadratic formula. So, \(x = [-1 ± sqrt((1)^2 - 4*2*(-4))] / 2*2\).

Step 3: Calculate the square root part

The part under the square is \(1 - 4*2*(-4)=17\). Therefore, the equation becomes: \(x = [-1 ± sqrt(17)] / 4\)

Step 4: Solve for x

Finally, this gives two possible solutions, \(x = [-1 + sqrt(17)] / 4 = 0.782\) and \(x = [-1 - sqrt(17)] / 4 = -2.282\). These are the roots of the quadratic equation.

Key Concepts

Quadratic Formula

The quadratic formula is a key tool in solving quadratic equations. It provides an efficient method to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula itself is written as:

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

This formula helps calculate the solutions (or roots) by substituting the coefficients \(a\), \(b\), and \(c\) from the quadratic equation into the formula. Remember that \(\pm\) in the formula means there are always potentially two solutions. These solutions arise because of the plus and minus operation in the formula, representing possible positions of the roots in relation to the graph of the equation (a parabola).
Furthermore, the term under the square root, \(b^2 - 4ac\) is called the discriminant. It determines the nature of the roots:

• If it's positive, there are two distinct real roots.
• If it's zero, there's exactly one real root.
• If it's negative, the roots are complex (not real numbers).

Roots of Equations

The roots of a quadratic equation are the values of \(x\) that make the equation true (i.e., zero). In terms of the graph of a quadratic equation, the roots represent the points where the parabola intersects the x-axis. These solutions can provide insights into the nature and direction of the parabola.
When we solve the equation \(2x^2 + x - 4 = 0\) using the quadratic formula, we found the roots to be approximately \(0.782\) and \(-2.282\). These roots tell us that the parabola crosses the x-axis at these points. It's important to note:

• The roots are often referred to as "zeros" because they represent the x-values where the equation results in zero.
• In some contexts, these solutions might be called solutions or x-intercepts, but they all guide us to the same understanding of the equation's graphical behavior.

Solving Quadratic Equations

Solving quadratic equations often requires a specific set of steps. Let's summarize these steps using our example equation, \(2x^2 + x - 4 = 0\):

• First, identify the coefficients \(a\), \(b\), and \(c\). In our case, \(a=2\), \(b=1\), and \(c=-4\).
• Next, substitute these values into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Simplify the expression under the square root (the discriminant), which for our equation is \(17\).
• Use the resolved formula \(x = \frac{-1 \pm \sqrt{17}}{4}\) to find the precise solutions.

The solutions \(0.782\) and \(-2.282\) show where the equation equals zero. Quadratic equations present a structured way of understanding mathematical relationships that involve squares. With practice, using the quadratic formula becomes a quick and reliable method for problem-solving.