Problem 30 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$-3 x^{2}-2 x-5=0$$

Step-by-Step Solution

Answer

The solutions are \[ x = -\frac{1}{3} \pm \frac{i \sqrt{14}}{3} \].

Step 1 - Identify coefficients

The given quadratic equation is ewline\[ -3x^2 - 2x - 5 = 0 \]ewlineIdentify the coefficients: \[ a = -3, \ b = -2, \ c = -5 \]

Step 2 - Write down the quadratic formula

The quadratic formula for solving \[ ax^2 + bx + c = 0 \] is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3 - Substitute coefficients into the formula

Substitute \[ a = -3, \ b = -2, \ c = -5 \] into the quadratic formula:\[ x = \frac{--2 \pm \sqrt{(-2)^2 - 4(-3)(-5)}}{2(-3)} \]

Step 4 - Simplify inside the square root

Compute inside the square root:\[ (-2)^2 - 4(-3)(-5) = 4 - 60 = -56 \]

Step 5 - Handle the square root of a negative number

Since the discriminant is negative, the equation has complex solutions. \[ \sqrt{-56} = i \sqrt{56} = i \sqrt{4 \cdot 14} = 2i\sqrt{14} \]

Step 6 - Complete the formula

Plug the simplified values back into the formula:\[ x = \frac{2 \pm 2i \sqrt{14}}{-6} \]Simplify the expression:\[ x = -\frac{1}{3} \pm \frac{i \sqrt{14}}{3} \]

Key Concepts

Complex Solutions

When solving quadratic equations using the quadratic formula, you may encounter complex solutions. These occur when the discriminant (the part inside the square root) is negative.
A negative discriminant means you have to deal with the square root of a negative number, which introduces the imaginary unit, denoted as \( i \).
The imaginary unit is defined such that \( i^2 = -1 \). For example, the square root of -56 can be expressed as \( i \sqrt{56} \).
This is useful in obtaining complex solutions, which are typically written in the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Discriminant

The discriminant is a specific part of the quadratic formula, found under the square root sign. The formula for the discriminant is:
\( b^2 - 4ac \) The discriminant determines the nature of the roots of the quadratic equation:

• If the discriminant is positive, there are two real and distinct solutions.


• If the discriminant is zero, there is exactly one real solution (also called a repeated or double root).


• If the discriminant is negative, there are no real solutions; instead, there are two complex solutions.

Understanding the discriminant is key when solving quadratic equations, as it helps you anticipate the type of solutions you will get.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, which means the highest power of the variable is squared. A standard form of a quadratic equation is:
\( ax^2 + bx + c = 0 \)
Here, \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents the variable to be solved for.
The solutions to these equations can be found using various methods, such as factoring, completing the square, or using the quadratic formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
The quadratic formula is particularly useful for any quadratic equation, as it provides a straightforward way to find the roots, even for equations that are difficult to factor or complete the square on.