Problem 102 - Algebra and Trigonometry

Question

Solve equation by the method of your choice. $$ 5 x^{2}=2 x-3 $$

Step-by-Step Solution

Answer

There are no real solutions to this equation because the discriminant (the value under the square root in the quadratic formula) is negative.

Step 1: Rearrange the Equation to Standard Form

Get all the terms on one side of the equation to put it in standard form.\n5x² - 2x + 3 = 0

Step 2: Identify a, b, c from the Equation

In a quadratic equation ax² + bx + c = 0, the coefficients are denoted as:\na = coefficient of x² = 5\nb = coefficient of x = -2\nc = constant term = 3

Step 3: Plug the Values into Quadratic Formula

The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a)\n Insert the values of a, b and c from the quadratic equation.\n x = [ -(-2) ± sqrt((-2)² - 4*5*3)] / (2*5)

Step 4: Simplify the Expression

Simplify the expression within the square root first, then the rest of the formula.\n x = [ 2 ± sqrt((4) - (60))] / (10)\n x = [ 2 ± sqrt(-56)] / (10)\n Note: Because the value under the square root is negative, there are no real solutions to this equation.

Key Concepts

Quadratic Formula

One of the most powerful tools for solving quadratic equations is the quadratic formula. It's a straightforward method that can solve any quadratic equation, regardless of how complex it appears. The formula is \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \] where \(a\), \(b\), and \(c\) are the coefficients of the equation \(ax^2 + bx + c = 0\).

To use this formula, you simply substitute the values of \(a\), \(b\), and \(c\) from your quadratic equation into the formula. The symbol \(\pm\) indicates that you'll get two solutions, one by adding the square root and another by subtracting it. However, if the value inside the square root, called the discriminant, is negative, you'll end up with complex numbers—which leads us to our next concept.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is expressed as \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants and \(x\) represents the unknown variable we are solving for. The values of \(a\), \(b\), and \(c\) dictate the shape and position of the parabolic graph that represents the equation.

The equation solved in our exercise was originally in the form of \(5x^2 = 2x - 3\). For using methods such as the quadratic formula, completing the square, or factoring, we first need to rearrange the equation to its standard form, as shown in the solution steps. Remember, \(a\) cannot be zero, as that would make the equation linear rather than quadratic.

Complex Numbers

In mathematics, complex numbers extend the concept of one-dimensional number line to a two-dimensional complex plane. A complex number is composed of a real part and an imaginary part and is generally written as \[ z = a + bi \] where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the defining property \(i^2 = -1\).

The existence of complex numbers allows for the solutions to certain quadratic equations that have no solution in the realm of real numbers. As you've observed, our equation produced a negative number under the square root, which means its solutions will be in the form of complex numbers.

No Real Solutions

There are instances when the discriminant (the value under the square root in the quadratic formula) is negative, leading to the square root of a negative number, which does not exist among the real numbers. In such cases, we say the quadratic equation has 'no real solutions.' Instead, the solutions are complex numbers, involving the imaginary unit \(i\).

When confronted with this situation, like in our exercise, we calculate the solutions using the quadratic formula and express the result as a pair of complex conjugates. This reflects an important principle in algebra: a quadratic equation always has two solutions, which may be real numbers, one real number (in a case of a repeated solution), or complex numbers.