Problem 8 - Precalculus : Building Concepts and Connections

Question

Use the quadratic formula to solve the equation. $$x^{2}+x-4=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{-1 + \sqrt{17}}{2}\) and \(x = \frac{-1 - \sqrt{17}}{2}\).

Step 1: Identify the coefficients a, b and c

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

Step 2: Substitute the coefficients into the quadratic formula

We can substitute the identified coefficients into the quadratic formula like this: \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). This becomes \(x = \frac{-1 \pm \sqrt{1+16}}{2}\).

Step 3: Solve the equation inside the square root

Calculate the value under the square root sign, which is \(1+16\) to get \(17\). So it becomes \(x = \frac{-1 \pm \sqrt{17}}{2}\).

Step 4: Final Solution

Solving further, we get two solutions: \(x = \frac{-1 + \sqrt{17}}{2}\) and \(x = \frac{-1 - \sqrt{17}}{2}\).

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental mathematical expression used to model parabolic shapes and relationships. It is generally in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are numbers known as coefficients, and \(x\) represents an unknown variable. The main feature of a quadratic equation is the presence of \(x^2\), which signifies that the equation is of the second degree.
Quadratic equations frequently appear in real-world scenarios such as physics, engineering, and finance, where they describe scenarios like projectile motion or the profit and cost analysis of businesses.
The equation \(x^2 + x - 4 = 0\) is a classic example, where the task is to find the value of \(x\) that satisfies the equation. These values of \(x\) are called the roots or solutions of the quadratic equation. To find them, mathematicians and students often rely on the quadratic formula.

Coefficients

Coefficients are the constants that multiply the variables or powers in polynomials. In the context of a quadratic equation like \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are the coefficients.
Let's explore their roles:

• \(a\) is the coefficient of \(x^2\). It determines the parabola's direction: if \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• \(b\) is the coefficient of \(x\). It affects the vertex's position and the symmetry of the parabola.
• \(c\) is the constant term. It tells us the point where the parabola intersects the y-axis.

In our example, the coefficients are \(a = 1\), \(b = 1\), and \(c = -4\). These values are essential for substituting into formulas and solving the equation.

Square Root

The square root is a mathematical operation used to find a number which, when multiplied by itself, gives the original number. It is symbolized as \(\sqrt{\cdot}\).
In the quadratic formula, the square root function plays a crucial role in determining the solutions, as it involves finding the square root of the discriminant \(b^2 - 4ac\).
This discriminant helps us determine:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is one real and repeated solution.
• If \(b^2 - 4ac < 0\), there are two complex solutions with imaginary components.

For our equation, the square root calculation involves \(b^2 - 4ac = 1 + 16 = 17\), which points to the existence of two real solutions. This step is vital as it directly impacts the nature of the solutions available for the equation.

Solution to Equations

Finding the solution to equations, especially quadratic ones, is about determining the values of \(x\) that satisfy the equation. With quadratic equations, the quadratic formula is a powerful tool that helps find the solutions efficiently.
The formula is written as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). By substituting the coefficients \(a\), \(b\), and \(c\) into this formula, the solutions emerge.
For our equation \(x^2 + x - 4 = 0\), we substitute the coefficients into the quadratic formula, leading to:

• \(x = \frac{-1 \pm \sqrt{17}}{2}\)

This provides two solutions:

• \(x = \frac{-1 + \sqrt{17}}{2}\)
• \(x = \frac{-1 - \sqrt{17}}{2}\)

Having two solutions means that the quadratic equation has two distinct points where the parabola crosses the x-axis, highlighting its symmetry and the elegance of its solutions.