Problem 70 - Algebra and Trigonometry

Question

Solve equation using the quadratic formula. $$ 5 x^{2}+x-2=0 $$

Step-by-Step Solution

Answer

The solution to the equation are \(x = (sqrt(41) - 1) / 10\) and \(x = -(sqrt(41) + 1) / 10\)

Step 1: Identify Coefficients

The quadratic equation is given as \(5x^2 + x - 2 = 0\). The coefficients are a=5, b=1, and c=-2.

Step 2: Plug Coefficients into Quadratic Formula

Substitute a, b, and c into the Quadratic formula: \(x = [-b ± sqrt(b^2 - 4ac)] / 2a\), which gives \(x = [-1 ± sqrt((1)^2 - 4*5*(-2))] / (2*5)\)

Step 3: Simplify under the square root

The square root part becomes sqrt(1 + 40) = sqrt(41). Therefore the equation simplifies to \(x = [-1 ± sqrt(41)] / 10\)

Step 4: Find the Roots

Finally, solve for the roots: \(x = [-1 + sqrt(41)] / 10\) and \(x = [-1 - sqrt(41)] / 10\). Thus the roots are \(x = (sqrt(41) - 1) / 10\) or \(x = -(sqrt(41) + 1) / 10\)

Key Concepts

Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable x with a non-zero coefficient for the term with the square of the variable. It has the standard form:

\[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the exercise, the quadratic equation is \( 5x^2 + x - 2 = 0 \), where \( a = 5 \), \( b = 1 \), and \( c = -2 \). Understanding this form is crucial since it's the foundation for solving quadratic equations using various algebraic methods, including the quadratic formula.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of x that satisfy the equation. There are several methods to solve such equations, including factoring, completing the square, and graphing. However, the most versatile and widely used method is the quadratic formula:

\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]

To apply the quadratic formula, you first determine the coefficients (a, b, and c). Then, you substitute these values into the formula. The symbol \( \pm \) indicates that there will typically be two solutions, known as the 'roots' of the equation. This method is particularly useful when the equation isn't easily factored or when dealing with non-integer coefficients, ensuring you always find the solutions for x.

Roots of a Quadratic

The roots of a quadratic equation are the x-values that make the equation equal to zero. These are also the points at which the parabola, the graph of a quadratic equation, intersects the x-axis. The quadratic formula provides these roots directly. In the example provided, the roots are calculated as follows:

\[ x = \frac{{-1 \pm \sqrt{{41}}}}{{10}} \]

The expression under the square root, \( b^2 - 4ac \), is called the discriminant and determines the nature of the roots. If it is positive, there are two distinct real roots, as in our exercise. If the discriminant equals zero, there is one real root, and if it's negative, there are two complex roots. This key concept helps students predict the number and type of solutions before even calculating them.

Algebraic Methods

There are multiple algebraic methods to solve quadratic equations. Each method has its advantages and situational benefits. For example, factoring is often the quickest method but only works easily when the equation is factorable into rational numbers. Completing the square is a method that can be used to solve any quadratic equation and forms the basis for the derivation of the quadratic formula. It involves creating a perfect square trinomial from the equation. The quadratic formula, used in the exercise, is derived from the process of completing the square and enables solving for roots without factoring. Understanding different algebraic methods empowers students to tackle a variety of quadratic equations effectively.