Problem 73 - Algebra and Trigonometry

Question

Solve equation using the quadratic formula. $$ x^{2}-6 x+10=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 3 + 1i \) and \( x = 3 - 1i \).

Step 1: Identify the coefficients

From the quadratic equation \( x^{2}-6x+10=0 \), coefficients of x are \( a = 1 \), \( b = -6 \), and \( c = 10 \).

Step 2: Substitute coefficients into the quadratic formula

Substitute \( a \), \( b \), and \( c \) into the quadratic formula \( \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \) yielding \( \frac{-(-6) \pm \sqrt{(-6)^{2}-4*1*10}}{2*1}\). This simplifies to \( \frac{6 \pm \sqrt{36-40}}{2}\) and then to \( \frac{6 \pm \sqrt{-4}}{2}\).

Step 3: Calculate the solutions

Since the square root of a negative number results in a complex number, the two solutions are: \( x = \frac{6 \pm \sqrt{-4}}{2}\). It can be simplified as \( x = 3 \pm 1i \) , which are two complex numbers. Hence the solutions are \( x = 3 + 1i \) and \( x = 3 - 1i \).

Key Concepts

Solving Quadratic Equations

Quadratic equations, which take the general form of \( ax^2 + bx + c = 0 \), are fundamental in algebra. The solutions to these equations are the values of \( x \) that make the equation true. To solve them, one of the most powerful tools we have is the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

When applying the quadratic formula, the first step is to correctly identify \( a \) (the coefficient in front of \( x^2 \) term), \( b \) (the coefficient in front of \( x \) term), and \( c \) (the constant term). Once you've substituted these values, simplify under the square root. If you encounter a negative number inside the square root, prepare to work with complex numbers, which leads to the next part: solving the equation when \( b^2 - 4ac < 0 \) and you're dealing with imaginary numbers.

Complex Numbers

Complex numbers arise in equations when we have a negative number under the square root, known as the discriminant. In our quadratic equation example, the discriminant is \( -4 \). Complex numbers consist of a real part and an imaginary part and are written in the form \( a + bi \), where \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).

In the given exercise solution, we are left with \( \sqrt{-4} \), which simplifies to \( 2i \), since \( 2^2 = 4 \) and by the property of the imaginary unit. Consequently, the solutions come out as \( x = 3 \pm 1i \). Understanding complex numbers is essential because they extend the real number system and have practical applications in various fields including engineering and physics.

Coefficients of Quadratic Equation

The quadratic equation's behavior and its graphical representation are determined by its coefficients \( a \) (quadratic coefficient), \( b \) (linear coefficient), and \( c \) (constant term). The coefficient \( a \) influences the opening direction and the width of the parabola. If \( a > 0 \) the parabola opens upwards, and if \( a < 0 \) it opens downwards.

The value of \( b \) affects the position of the vertex along the horizontal axis, changing how the graph shifts left or right. Meanwhile, \( c \) represents the \( y \) -intercept. It is the point where the graph crosses the \( y \) -axis. When solving quadratic equations, it’s crucial to accurately identify these coefficients, to not only find the roots but also to understand the nature of the graph and the function's behavior.