Problem 79 - College Algebra Essentials

Question

In Exercises \(75-82,\) compute the discriminant. Then determine the number and type of solutions for the given equation. $$ x^{2}-2 x+1=0 $$

Step-by-Step Solution

Answer

The discriminant is 0. Hence, the equation has exactly one real solution.

Step 1: Identifying a, b, c values

For a quadratic equation in the form \(ax^{2} + bx + c = 0\), identify the coefficients \(a, b, c\). Here, \(a=1, b = -2, c = 1 \)

Step 2: Compute the discriminant

The discriminant is given by \(b^{2}-4ac\). Subsitute \(a, b, c\) values into the discriminant formula: \((-2)^{2} - 4*1*1 = 4 - 4 = 0\). The discriminant is 0.

Step 3: Identify the type and number of solutions

As the discriminant is 0, the quadratic equation has exactly one real solution.

Key Concepts

Quadratic Equation Solutions

A quadratic equation is a polynomial equation of degree two. It typically takes the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants. Solving a quadratic equation means finding the values of \(x\) that make the equation true. These values are called the solutions or roots of the equation. The solutions to a quadratic equation can be found using several methods:

• Factoring, if the equation can easily be factored into two binomial expressions.
• Completing the square, which involves rearranging the equation and making a perfect square trinomial.
• The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which is derived from completing the square and works for any quadratic equation.

Understanding quadratic equation solutions involves recognizing that the number and type of solutions depend on the discriminant, \(b^2 - 4ac\), found under the square root sign in the quadratic formula.

Discriminant Formula

The discriminant is a crucial component in determining the nature and number of solutions to a quadratic equation. Given a quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant is defined as \(\Delta = b^2 - 4ac\). Calculating the discriminant involves simply substituting the coefficients \(a, b,\) and \(c\) into this formula. The value of the discriminant tells you:

• If \(\Delta > 0\), the quadratic equation has two distinct real solutions. This occurs because the square root of a positive number is real and has two values (positive and negative).
• If \(\Delta = 0\), there is exactly one real solution. The roots are the same, hence called a repeated root.
• If \(\Delta < 0\), the equation has no real solutions but has two complex solutions. This is because the square root of a negative number involves imaginary numbers.

By using the discriminant formula, one can quickly assess the characteristics of the solutions without fully solving the equation.

Real and Complex Solutions

The solutions of a quadratic equation are greatly influenced by the value of the discriminant, \(\Delta = b^2 - 4ac\). They can be either real or complex, depending on whether \(\Delta\) is positive, zero, or negative. **Real Solutions**

• When \(\Delta > 0\), the equation has two different real solutions. This means you find two distinct x-values that satisfy the equation.
• If \(\Delta = 0\), the equation has one real solution. In this case, the parabola touches the x-axis at exactly one point, indicating a repeated or double root.

**Complex Solutions**

• When \(\Delta < 0\), the equation does not intersect the x-axis, meaning no real x-values solve the equation. Instead, the solutions are complex (or imaginary) and come in conjugate pairs, e.g., \(a + bi\) and \(a - bi\).

Understanding whether solutions are real or complex informs us about the behavior of the quadratic function's graph and gives insight into possible applications in various fields such as engineering and physics.