Problem 34 - Algebra A Combined Function

Question

Use the quadratic formula to solve each equation. These equations have real solutions and complex, but not real, solutions. $$ x(7 x+1)=2 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-1 + \sqrt{57}}{14}\) and \(x = \frac{-1 - \sqrt{57}}{14}\).

Step 1: Expand the Equation

First, expand the given equation by distributing the term \(x\). This gives: \(x(7x + 1) = 7x^2 + x\). So, the equation becomes \(7x^2 + x = 2\).

Step 2: Rearrange into Standard Form

Bring all terms to one side of the equation to form a standard quadratic equation \(ax^2 + bx + c = 0\). Subtract 2 from both sides to obtain: \(7x^2 + x - 2 = 0\).

Step 3: Identify Coefficients

Identify the coefficients in the quadratic equation \(7x^2 + x - 2 = 0\). Here, \(a = 7\), \(b = 1\), and \(c = -2\).

Step 4: Apply the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the identified coefficients into this formula: \(x = \frac{-(1) \pm \sqrt{(1)^2 - 4 \cdot 7 \cdot (-2)}}{2 \cdot 7}\).

Step 5: Simplify the Discriminant

Calculate the discriminant \(b^2 - 4ac\): \((1)^2 - 4 \times 7 \times (-2) = 1 + 56 = 57\).

Step 6: Simplify the Expression

Substitute the discriminant back into the quadratic formula: \(x = \frac{-1 \pm \sqrt{57}}{14}\).

Step 7: Solve for x

Evaluate \(x\) using the \(\pm\) operations: \(x = \frac{-1 + \sqrt{57}}{14}\) and \(x = \frac{-1 - \sqrt{57}}{14}\).

Key Concepts

Real Solutions

The term "real solutions" in the context of quadratic equations refers to the roots of the equation that can be found on the number line. These numbers are non-imaginary and tangible without involving any complex numbers. Real solutions occur when the discriminant, a key component found within the quadratic formula, is zero or positive.
The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), plays a vital role here. The part under the square root sign, \(b^2 - 4ac\), determines the nature of the roots: whether they are real or complex.

• When \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is one real solution (or a repeated real root).
• And if \(b^2 - 4ac < 0\), the roots become complex or imaginary, which we will discuss next.

Real solutions are crucial in practical problems where results need to be meaningful in the physical world, like measurements or predictions.

Complex Solutions

Complex solutions arise when the quadratic equation's discriminant is negative. In such cases, the roots of the equation involve imaginary numbers, which are numbers that include \(i\), the square root of -1.
When the discriminant \(b^2 - 4ac < 0\), it indicates that the square root of a negative number is involved in the solution, resulting in complex numbers of the form \(a + bi\) and \(a - bi\).

• These solutions are not on the real number line because they extend into a plane known as the complex plane.
• Despite this, complex solutions play an essential role, particularly in advanced fields such as engineering and physics.

Understanding the concept of complex solutions helps in grasping multifaceted problems that involve oscillations, wave mechanics, and signal processing.

Discriminant

The discriminant, denoted as \(b^2 - 4ac\), is a critical component in determining the nature of the solutions of a quadratic equation. In essence, it provides insight into whether the solutions are real or complex without having to find the roots themselves.
Evaluating the value of the discriminant quickly informs us about the nature of the solutions:

• If \(b^2 - 4ac > 0\), expect two distinct real roots.
• If \(b^2 - 4ac = 0\), there will be exactly one real root, known as a repeated or double root.
• If \(b^2 - 4ac < 0\), the solutions will be complex.

In our specific example, with a discriminant of 57, we know that the solutions are real, as 57 is a positive number. This understanding allows for efficient problem solving in mathematics and real-world scenarios.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\). This format is ideal for easy identification of the coefficients and simplification into the quadratic formula.
In our example, the equation was initially given as \(x(7x + 1) = 2\). To transform it into standard form, some steps are necessary:

• First, expand any factored expressions, as was done to find \(7x^2 + x = 2\).
• Next, reorganize the equation to set it equal to zero, resulting in \(7x^2 + x - 2 = 0\).

Now, the coefficients \(a\), \(b\), and \(c\) can be easily identified, simplifying the application of the quadratic formula or any other method required for solving the equation. This structure is fundamental to any quadratic problem-solving approach.