Problem 83 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ x^{2}-2 x=-1 $$

Step-by-Step Solution

Answer

The equation has one real solution: \(x = 1\).

Step 1: Arrange the Equation

The given equation is \(x^2 - 2x = -1\). To express it in the standard form \(ax^2 + bx + c = 0\), we need to move all terms to one side. Add 1 to both sides to obtain:\[x^2 - 2x + 1 = 0\]This is now in the standard form with \(a = 1\), \(b = -2\), and \(c = 1\).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) is given by the formula \(b^2 - 4ac\). Substituting the values from the standard form, we get:\[\Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0\]The discriminant is 0.

Step 3: Determine the Number of Real Solutions

Since the discriminant \(\Delta = 0\), the equation has exactly one real solution. This means the roots are real and repeated.

Step 4: Solve the Equation

Since the discriminant is 0, the solution can be found using the formula for roots of a quadratic equation, which is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute \(a = 1\), \(b = -2\), and \(c = 1\):\[x = \frac{-(-2) \pm \sqrt{0}}{2(1)} = \frac{2}{2} = 1\]Thus, the solution is \(x = 1\).

Key Concepts

Discriminant

The discriminant is an important part of a quadratic equation. It helps determine the nature of the roots. For a quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is calculated using the formula:

• \(\Delta = b^2 - 4ac\)

The value of the discriminant tells us how many real solutions an equation has.
If \(\Delta > 0\), the equation has two distinct real roots. If \(\Delta = 0\), there is exactly one real root.
And if \(\Delta < 0\), there are no real roots, only complex ones. In this exercise, the discriminant is 0, indicating one real and repeated root.

Standard Form

Quadratic equations are usually expressed in the standard form which is essential for solving them easily. The standard form is:

• \(ax^2 + bx + c = 0\)

In this format, \(a\), \(b\), and \(c\) are constants, with \(aeq0\).
Converting an equation to this form means moving all terms to one side of the equation and organizing them by degree from highest to lowest.
For example, the equation \(x^2-2x=-1\) was rearranged to \(x^2 - 2x + 1 = 0\), allowing further analysis and solving.

Real Solutions

Real solutions refer to the actual values of \(x\) that satisfy the quadratic equation. They are the points where the graph of the quadratic touches or intersects the x-axis.
The discriminant helps determine the number of real solutions:

• A positive discriminant \(>0\) indicates two distinct real solutions.
• A zero discriminant \(=0\) indicates one real double root.
• A negative discriminant \(<0\) indicates no real solutions.

In this problem, we found that the discriminant \(\Delta = 0\).
Thus, there is exactly one real solution meaning the parabola just touches the x-axis.

Roots of Quadratic Equation

The roots or solutions of a quadratic equation are the values of \(x\) for which the equation equals zero. These roots can be found using the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula can be applied directly when the equation is in standard form.
For our problem, after solving, we found that \(x = 1\) is the root.
The symbol "\(\pm\)" in the formula indicates that there could be two potential values for \(x\), depending on the sign of the square root.
However, since the discriminant here was zero, both potential roots are actually the same, confirming the single real and repeated root \(x = 1\).