Problem 39 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$7 x^{2}+2 x-1=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(7x^2 + 2x - 1 =0\) are \(x = -\frac{1}{7} + \frac{2\sqrt{2}}{7}\) and \(x = -\frac{1}{7} - \frac{2\sqrt{2}}{7}\).

Step 1: Identify a, b and c values

Compare the given quadratic equation \(7x^2 + 2x - 1 =0\) with the standard form. For this equation, the coefficients are \(a = 7\), \(b = 2\), and \(c = -1\).

Step 2: Plug in the values into the quadratic formula

We substitute \(a\), \(b\), and \(c\) into the quadratic formula. This gives us: \(x = \frac{-2 \pm \sqrt{2^2-4*7*(-1)}}{2*7}\)

Step 3: Simplify the expression under the square root

This step requires us to calculate the discriminant, \(b^2-4ac\). We obtain: \(x = \frac{-2 \pm \sqrt{4+28}}{14}\)

Step 4: Simplify the entire equation

We find: \(x = \frac{-2 \pm \sqrt{32}}{14} = \frac{-2 \pm 4\sqrt{2}}{14}\).

Step 5: Simplify the solutions

Finally, simplify to find: \(x = -\frac{1}{7} \pm \frac{2\sqrt{2}}{7}\)

Key Concepts

Discriminant

In a quadratic equation, the discriminant plays a crucial role in determining the nature of the solutions. It is expressed as \(b^2 - 4ac\) in the quadratic formula, which is \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \].
Understanding the discriminant is very important as it reveals whether the roots of the equation are real or complex numbers.

• If the discriminant is positive, \(b^2-4ac > 0\), the equation has two distinct real solutions.
• If it is zero, \(b^2-4ac = 0\), the equation has exactly one real solution or a repeated real solution.
• If the discriminant is negative, \(b^2-4ac < 0\), the equation has two complex solutions.

In the given exercise, the discriminant calculated as \(4 + 28\) equals 32, indicating two distinct real solutions.

Quadratic Equation

The quadratic equation is a polynomial equation of the second degree, typically written as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) represent constants with \(a eq 0\). This form of equation is crucial as it often arises in various mathematical scenarios and real-world applications, such as physics and engineering.
Quadratics are solved using different methods:

• Factoring (if possible)
• Completing the square
• Quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In the original exercise, the quadratic formula method is used, demonstrating a comprehensive way to handle problems that might be tricky to factor or complete square.

Real and Complex Solutions

Solutions to quadratic equations can be either real or complex numbers. As mentioned earlier, the discriminant helps us determine the nature of these solutions.

**Real Solutions**: If the discriminant is positive or zero, the roots are real. They appear as distinct or repeated real numbers, respectively. Real solutions are often desired, as they can be directly applied to real-world problems.
**Complex Solutions**: When the discriminant is negative, the roots involve imaginary numbers, leading to complex solutions. Complex numbers combine a real part and an imaginary part, expressed as \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\).
In cases where mathematical models require analysis in terms of all possible solutions, such as in electrical engineering and fluid dynamics, complex solutions prove invaluable.