Problem 22 - Introductory Algebra for College Students

Question

$$3 x^{2}=6 x-14$$

Step-by-Step Solution

Answer

The roots of the given equation are not real.

Step 1 - Rewrite the equation in standard form

To rewrite the equation in standard form, all terms need to be on one side of the equation: \(3x^2 - 6x + 14 = 0\).

Step 2 - Identify coefficients

In the equation \(3x^2 - 6x + 14 = 0\), \(a = 3\), \(b = -6\), and \(c = 14\). These coefficients will be used in the quadratic formula.

Step 3 - Apply the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting \(a = 3\), \(b = -6\), and \(c = 14\) gives \(x = \frac{6 \pm \sqrt{(-6)^2 - 4*3*14}}{2*3}\).

Step 4 - Simplify the results

After simplifying, \(x = \frac{6 \pm \sqrt{36 - 168}}{6}\). As \(\sqrt{36 - 168}\) is not a real number, the roots of the given equation are not real numbers.

Key Concepts

Quadratic Formula

In algebra, the quadratic formula is an essential tool for solving quadratic equations. These equations have the general form: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants. The quadratic formula allows you to find the values of \( x \) that satisfy the equation.

• The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• The part under the square root sign, \( b^2 - 4ac \), is known as the discriminant.
• If the discriminant is positive, you get two distinct real solutions. If it's zero, there is one repeated real solution. If it's negative, the solutions are complex numbers.

The quadratic formula is widely used due to its ability to solve any quadratic equation, whether the roots are real or complex. Remember, the formula handles complex numbers when required.

Complex Numbers

Complex numbers provide solutions when a quadratic equation does not have real roots. They arise when dealing with the square root of a negative number.

• A complex number is of the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.
• The imaginary unit \( i \) is defined as \( i^2 = -1 \).

In quadratic equations, you encounter complex numbers when the quadratic formula results in a negative discriminant.For example, if you have: \[ \sqrt{b^2 - 4ac} \] in the formula and \( b^2 - 4ac \) is negative, then the roots are complex.By using complex numbers, you can work with equations that don't intersect the x-axis on a graph, providing a complete set of solutions.

Standard Form

Writing a quadratic equation in standard form is the first step to solving it using the quadratic formula. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).

• It is crucial to arrange the equation like this before applying the quadratic formula.
• This form makes it easy to identify the coefficients \( a \), \( b \), and \( c \), which are necessary for the formula.

Converting any given quadratic equation into this form might involve moving terms from one side of the equation to the other, just as in the provided exercise, where starting with \( 3x^2=6x-14 \) was rearranged to \( 3x^2 - 6x + 14 = 0 \). This form is standardized so mathematicians can tackle quadratic equations consistently and accurately.