Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation using any convenient method. $$x^{2}-2 x=-\frac{13}{4}$$

Step-by-Step Solution

Answer

The equation \(x^2-2x - \frac{13}{4} = 0\) has no real solution since the discriminant(b^2 - 4ac) results in a negative number.

Step 1: Write the equation in the standard form

Rearrange the equation to standard form, which is \(ax^2 + bx + c = 0\). So, αdd \(\frac{13}{4}\) to both sides: \(x^2 - 2x + \frac{13}{4} = 0\)

Step 2: Apply the quadratic formula

The quadratic formula is given by \(x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\). Here a=1, b=-2, c=\(\frac{13}{4}\). By substituting these values into the quadratic formula, we get: \(x = 1 ± \sqrt{1 - (\frac{13}{1})}\)

Step 3: Simplify the roots

Simplifying the expression inside the square root we get -12, but the square root of a negative number is undefined in the real number system. Therefore, there is no real solution to the equation.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving equations of the form \(ax^2 + bx + c = 0\), known as quadratic equations. This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), allows us to find the roots of any quadratic equation by substituting in the coefficients \(a\), \(b\), and \(c\).

Understanding how to utilize this formula is vital. After you've written the equation in standard form and identified your coefficients, you simply plug them into the formula. In some cases, the term under the square root, known as the discriminant, can be negative. This leads us directly into the realm of complex numbers, which we will explore in the next section. Keep in mind that simplifying the quadratic formula requires careful attention to arithmetic operations and often involves the simplification of square roots as well.

Complex Numbers

Complex numbers extend the idea of the traditional number system to include the square roots of negative numbers. The basic unit of complex numbers is \(i\), where \(i^2 = -1\).

When you come across a negative discriminant in the quadratic formula, the roots of the equation will be complex numbers. Using our current example, the discriminant is \(b^2 - 4ac\), which gives us -12. To find the square root of -12, we write it as \(\sqrt{-12} = \(i\)\(\sqrt{12}\)\) to indicate the presence of a complex number. This is a fundamental concept in algebra because it broadens the scope of solvable equations, ensuring that every quadratic equation has a solution, be it real or complex.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is crucial for using the quadratic formula effectively. The standard form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known coefficients and \(x\) is the variable we solve for.

A proper arrangement into the standard form can often reveal information about the equation, such as symmetry of its graph and the nature of its roots, which can be real or complex. Additionally, the coefficient \(a\) influences the parabola's opening direction, while \(b\) and \(c\) affect the location of its vertex and the equation's \(y\)-intercept.

Simplifying Square Roots

Simplifying square roots involves expressing a square root in the simplest possible form. It's a matter of breaking down the number under the root into its prime factors and then pairing them to simplify. Numbers that are perfect squares, such as 4, 9, 16, simplify very easily.

However, when dealing with a negative number under the square root, as in our quadratic exercise, you cannot simplify it further with real numbers. This is where a basic understanding of complex numbers becomes essential. The principles of simplifying square roots help us to make sense of the discriminant in the quadratic formula and, hence, determine the nature of the roots of a quadratic equation.