Problem 60 - College Algebra

Question

Solve each equation in Exercises \(55-64\) using the quadratic formula. $$ 5 x^{2}+x-2=0 $$

Step-by-Step Solution

Answer

The solutions of the given quadratic equation are \( x_1 = \frac{-1 + \sqrt{41}}{10} \) and \( x_2 = \frac{-1 - \sqrt{41}}{10} \).

Step 1: Identify the coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the given quadratic equation. Here, \(a=5\), \(b=1\), and \(c=-2\) are the coefficients.

Step 2: Substitute the coefficients into the quadratic formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula. This gives \(x = \frac{-1 \pm \sqrt{1^2 - 4*5*-2}}{2*5}\)

Step 3: Simplify the equations

Simplify the equations to get the roots. The simplified equations are \(x = \frac{-1 \pm \sqrt{1 + 40}}{10}\) which further simplifies to \(x = \frac{-1 \pm \sqrt{41}}{10}\)

Step 4: Solve the equations

Now, solve the two simplified equations individually to obtain the two possible solutions. Therefore, the solutions are \( x_1 = \frac{-1 + \sqrt{41}}{10} \) and \( x_2 = \frac{-1 - \sqrt{41}}{10} \)

Key Concepts

Understanding Quadratic Equations

A quadratic equation is an expression that involves a polynomial degree of two, usually written in the standard form: \( ax^2 + bx + c = 0 \). The crucial aspect here is that your variable \( x \) is squared. It means the highest power of \( x \) is two, hence the term "quadratic." This type of equation forms a parabola when graphed on a coordinate plane.
To solve a quadratic equation, we often seek the values of \( x \) which satisfy or "solve" the equation, meaning that when these values are plugged into the equation, it results in a true statement (usually equating to zero). Learning how to manipulate and solve these equations is fundamental in algebra and various applications in science and engineering, where system dynamics or trajectories can frequently be described by quadratic relationships.

Identifying Coefficients

In a quadratic equation such as \( ax^2 + bx + c = 0 \), the letters \( a \), \( b \), and \( c \) are known as coefficients. They are constants that give the specific parabola its shape and position on a graph. Understanding how to identify and work with these coefficients is key to solving quadratic equations efficiently.

• \( a \) is the coefficient of \( x^2 \), and it determines the "width" and "direction" of the parabola. If \( a \) is positive, the parabola opens upwards, if negative, it opens downwards.
• \( b \) is the coefficient of \( x \), influencing the "tilt" or orientation of the parabola across the y-axis.
• \( c \) is the constant term, which shows where the parabola intersects the y-axis.

In the equation given \( 5x^2 + x - 2 = 0 \), \( a=5 \), \( b=1 \), and \( c=-2 \). Correctly identifying these coefficients enables you to use the quadratic formula to find the equation's solutions.

Solutions to Quadratic Equations Using the Quadratic Formula

The quadratic formula is an essential tool that provides a direct way to find the solutions or "roots" of a quadratic equation. The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This equation may seem complex at first, but breaking it down makes the process straightforward. The \( \pm \) sign indicates there can be two solutions, which correspond to the potential intersection points of the parabola with the x-axis. Here’s how each part works:

• \(-b\) is simply the negative of the coefficient \( b \).
• \( \sqrt{b^2 - 4ac} \) is under a square root, called the discriminant. It tells us about the nature of the roots:
  • If positive, the quadratic has two distinct real solutions.
  • If zero, there is exactly one real solution, called a "repeated" or "double" root.
  • If negative, the solutions are complex or imaginary.
• Finally, everything is divided by \( 2a \).

For \( 5x^2 + x - 2 = 0 \), substituting \( a=5 \), \( b=1 \), \( c=-2 \), gives us:\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 5 \cdot -2}}{2 \cdot 5} \]Simplified to:\[ x = \frac{-1 \pm \sqrt{41}}{10} \]Thus, the solutions are \( x_1 = \frac{-1 + \sqrt{41}}{10} \) and \( x_2 = \frac{-1 - \sqrt{41}}{10} \), describing the exact values where the function crosses the x-axis.