Problem 8 - Algebra 2

Question

Find the exact solutions by using the Quadratic Formula. \(4 x^{2}+20 x+25=-2\)

Step-by-Step Solution

Answer

Solutions are \( x = -\frac{5}{2} + \frac{\sqrt{2}i}{2} \) and \( x = -\frac{5}{2} - \frac{\sqrt{2}i}{2} \).

Step 1: Set Equation to Standard Form

To apply the quadratic formula, the equation must be in the form \( ax^2 + bx + c = 0 \). Start by adding 2 to both sides to obtain \(4x^2 + 20x + 27 = 0\).

Step 2: Identify Coefficients

In the equation \(4x^2 + 20x + 27 = 0\), identify the coefficients as follows: \(a = 4\), \(b = 20\), and \(c = 27\).

Step 3: Write Down the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula will be used to find the roots of the quadratic equation.

Step 4: Calculate the Discriminant

The discriminant is \( b^2 - 4ac \). Substitute the values to get \( 20^2 - 4 \times 4 \times 27 \), which simplifies to \(400 - 432 = -32\).

Step 5: Evaluate the Square Root of the Discriminant

Because the discriminant \(-32\) is negative, the solutions will be complex numbers. The square root of \(-32\) is \(\sqrt{32}i\), where \( i \) is the imaginary unit, and \( \sqrt{32} = 4\sqrt{2} \).

Step 6: Substitute Values into the Quadratic Formula

Plug the values into the quadratic formula: \[ x = \frac{-20 \pm 4\sqrt{2}i}{8} \].

Step 7: Simplify the Expression

Simplify the expression by dividing both the numerator and each term by 8: \( x = \frac{-20}{8} \pm \frac{4\sqrt{2}i}{8} \), which further reduces to \( x = -\frac{5}{2} \pm \frac{\sqrt{2}i}{2} \).

Key Concepts

Complex Numbers

When solving quadratic equations, especially when the discriminant is negative, you may encounter complex numbers. A complex number has a real part and an imaginary part. The standard form of a complex number is written as \(a + bi\), where:

• \(a\) represents the real part.
• \(b\) represents the imaginary part, where \(i\) is the imaginary unit.

In the given exercise, the solutions involve complex numbers because the discriminant was negative. Thus, the equation has no real solutions. Instead, it results in solutions of the form \(-\frac{5}{2} \pm \frac{\sqrt{2}i}{2}\), blending both real and imaginary components. Learning to work with complex numbers is key when diving into more advanced areas of mathematics.

Discriminant

The discriminant is a specific part of the quadratic formula that you calculate before solving the equation. It is denoted as \(b^2 - 4ac\). The value of the discriminant can tell you a lot about the nature of the solutions:

• If the discriminant is positive, there are two distinct real solutions.
• If it equals zero, there is one real solution, often called a repeated or double root.
• If it's negative, the solutions are not real but complex numbers, as it is in this case.

In this exercise, the discriminant was calculated as \(-32\), which is negative. Therefore, the solutions are complex numbers, indicating that the quadratic equation does not cross the x-axis and has no real roots.

Standard Form of a Quadratic Equation

Before applying the quadratic formula, it’s important to have the equation in the correct form. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where:

• \(a\), \(b\), and \(c\) are coefficients and \(a eq 0\).

The problem requires us to first adjust the given equation to this standard form. In this instance, moving -2 to the other side of the equation transforms \(4x^2 + 20x + 25 = -2\) into \(4x^2 + 20x + 27 = 0\).

• This step ensures that we can easily identify the coefficients \(a, b,\) and \(c\) for use in the quadratic formula.

Getting the equation into the standard form is crucial as it sets the stage for accurate calculation of the discriminant and eventual solution.

Imaginary Unit

Understanding the imaginary unit \(i\) is essential when dealing with complex numbers. The imaginary unit is defined such that \(i^2 = -1\). This property helps simplify the square roots of negative numbers into an understandable form.

• When you encounter a negative under the square root, the result will involve \(i\).
• For example, \(\sqrt{-1} = i\).

In the quadratic formula, when you have a negative discriminant, say \(-32\), it is interpreted as \(\sqrt{32}i\). The \(i\) signifies that you’re working with an imaginary component. Always remind yourself that \(i\) represents \(\sqrt{-1}\), which is a stepping stone to solving many algebraic problems involving complex solutions.