Problem 79 - Algebra and Trigonometry

Question

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 of the equation \(x^{2} - 2x + 1=0\) is 0. Thus, there is exactly one real solution.

Step1: Identify coefficients

For the given quadratic equation \(x^{2} - 2x + 1=0\), identify the coefficients of x^2, x, and the constant term. Here, a=1, b=-2, and c=1.

Step 2: Compute the discriminant

The discriminant (D) of a quadratic equation with coefficients a, b, c is given by \(D = b^{2} - 4ac\). Substituting a=1, b=-2, and c=1 into this yields \(D = (-2)^{2} - 4(1)*(1) = 4 - 4 = 0\).

Step 3: Determine the number and type of solutions

Since the discriminant D=0, there is exactly one real root for the given quadratic equation. This tells us that the equation \(x^{2} - 2x + 1=0\) has exactly one real solution.

Key Concepts

Quadratic Equation Solutions

Understanding the solutions to quadratic equations is essential for grasping higher-level algebra. A quadratic equation typically takes the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The solutions, also known as roots, are the values of \(x\) that satisfy the equation.

The nature and number of solutions are determined by the discriminant, which we compute using the formula \(D = b^2 - 4ac\). Depending on the value of the discriminant, there are three possible scenarios for the solutions of a quadratic equation:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution.
• If \(D < 0\), there are no real solutions, but two complex solutions.

These solutions can be found using the quadratic formula \(x = \frac{{-b \pm \sqrt{{D}}}}{{2a}}\), or by factoring the equation, completing the square, or using graphical methods. The method of solution can vary based on the equation's complexity and the given parameters.

Solve Quadratic Equations

To solve quadratic equations efficiently, one must be familiar with different methods and when to use them. Among these methods, the most universal is the quadratic formula, which works for any quadratic equation. Regardless of the discriminant, you can apply the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) to find the solutions.

Another common method is factoring, where you express the quadratic expression as a product of two binomials. Factoring is quick and effective when applicable, but it requires the quadratic to be factorable over the integers. Completing the square is another technique, involving rewriting the equation in the form of \(\left(x-h\right)^2 = k\), which makes it straightforward to solve for \(x\). Lastly, graphing the function \(y = ax^2 + bx + c\) allows you to visually identify the roots as the points where the graph intersects the \(x\)-axis. Each method has its advantages, and understanding when to use each is key to solving quadratic equations efficiently.

Discriminant Computation

The discriminant is a critical component in understanding quadratic equations as it determines the nature and number of roots without solving the equation. The discriminant computation formula is \(D = b^2 - 4ac\), where \(b\), \(a\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

To compute the discriminant, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Substitute these values into the discriminant formula.
• Calculate the value of \(D\) to determine the roots.

For instance, in the equation presented, \(x^2 - 2x + 1 = 0\), the coefficients are \(a = 1\), \(b = -2\), and \(c = 1\). Using the discriminant formula, we compute \(D = (-2)^2 - 4(1)(1) = 0\). This computation tells us that the equation has exactly one real solution, as a zero discriminant indicates a perfect square. It's important to understand the relationship between the discriminant and the quadratic equation's roots, as it can save time and effort when solving complex problems.