Problem 89 - Elementary Algebra

Question

Solve using the quadratic formula. $$ (2 x+3)(2 x-3)-5(x 2+1)=-9 $$

Step-by-Step Solution

Answer

The solutions are \(x = \sqrt{5}i\) and \(x = -\sqrt{5}i\).

Step 1: Expand Both Sides of the Equation

Begin by expanding the left-hand side of the equation. First, expand \((2x + 3)(2x - 3) = (2x)^2 - (3)^2 = 4x^2 - 9\).Next, expand \(-5(x^2 + 1) = -5x^2 - 5\).This makes the expression: \[(4x^2 - 9) - (5x^2 + 5) = -9\].

Step 2: Simplify the Equation

Combine like terms in the equation.\[4x^2 - 5x^2 - 9 - 5 = -9\].This simplifies to:\[-x^2 - 14 = -9\].

Step 3: Rearrange into Standard Quadratic Form

Move all terms to one side to form a standard quadratic equation \(ax^2 + bx + c = 0\):\[-x^2 - 14 + 9 = 0\].This simplifies to:\[-x^2 - 5 = 0\].Which can be rearranged to:\[x^2 + 5 = 0\].

Step 4: Solve Using the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).Identify the coefficients from the equation \(x^2 + 0x + 5 = 0\):\(a = 1, b = 0, c = 5\).Substitute these into the formula:\[x = \frac{-0 \pm \sqrt{0^2 - 4 \times 1 \times 5}}{2 \times 1} = \frac{\pm \sqrt{-20}}{2}\].

Step 5: Simplify the Solution

Calculate \(\sqrt{-20}\) to express it in terms of \(i\) (where \(i = \sqrt{-1}\)):\(\sqrt{-20} = \sqrt{4 \times 5} \cdot \sqrt{-1} = 2\sqrt{5}i\).Substitute back into the formula:\[x = \frac{\pm 2\sqrt{5}i}{2} = \pm \sqrt{5}i\].Thus, the solutions are \(x = \sqrt{5}i\) and \(x = -\sqrt{5}i\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). It's highlighted by its general formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula is helpful because it provides a way to solve quadratic equations even when they cannot be easily factored. Each part of the formula helps us to determine the roots:

• \(b^2 - 4ac\) is known as the discriminant. It tells us about the nature of the roots.
• If the discriminant is positive, there are two real roots.
• If the discriminant is zero, there is one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots.

When utilizing it, always pay attention to the coefficients \(a\), \(b\), and \(c\). These values must be identified correctly to substitute into the formula and find the correct solutions.

Complex Numbers

Complex numbers often arise in solutions to quadratic equations, particularly when the discriminant is negative. A complex number consists of two parts: a real part and an imaginary part, represented as \(a + bi\), where:

• \(a\) is the real part.
• \(b\) is the imaginary part.
• \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\).

In the context of solving the quadratic equation \(x^2 + 5 = 0\), the presence of a negative discriminant \((-20)\) leads to imaginary roots. Applying the imaginary unit \(i\), the roots are expressed as \(\sqrt{5}i\) and \(-\sqrt{5}i\), indicating purely imaginary solutions. Understanding and handling imaginary numbers is crucial for solving such equations effectively, especially since real-world applications can involve complex numbers as well.

Algebraic Expansion

Algebraic expansion is the process of multiplying out expressions such as products of polynomials. In this exercise, expansion involves distributing terms and simplifying the expression:

• For \((2x + 3)(2x - 3)\), the difference of squares formula is used: \((a + b)(a - b) = a^2 - b^2\).
• This simplifies the expression to \(4x^2 - 9\).
• Similarly, \(-5(x^2 + 1)\) expands to \(-5x^2 - 5\).

Combining these expanded terms forms a quadratic equation. Mastering algebraic expansion helps in simplifying equations and revealing the quadratic form needed for further solving.

Solving Equations

Solving equations involves finding the value(s) of variables that make the equation true. For a quadratic equation, this often involves manipulating and simplifying the expression to use the quadratic formula or other methods like factoring or completing the square.The process involves:

• Expanding and simplifying terms to bring the equation into a standard form \( ax^2 + bx + c = 0 \).
• Moving all terms to one side to ensure the equation is set to zero.
• Applying an appropriate solution method, in this case, the quadratic formula, to find the roots.

Each solving step involves careful attention to details, from the expansion of terms to ensuring correct arithmetic operations. For quadratic equations with no real solutions, complex roots are expected, highlighting the importance of understanding both algebra and complex number systems.