91Ó°ÊÓ

The standard form of a quadratic equation is crucial in solving it effectively. It is generally written as:
\( ax^2 + bx + c = 0 \)
Here, \(a\), \(b\), and \(c\) are constants. To solve a quadratic equation using any method, such as factoring or the quadratic formula, you must first rewrite it in this form. In the exercise, the given equation was \( x^{2}+1=x \). By moving all the terms to one side, we can write it as:
\( x^2 - x + 1 = 0 \)
This is now in standard form, where \(a = 1\), \(b = -1\), and \(c = 1\). Putting the equation in standard form helps identify these coefficients, which is necessary for applying the quadratic formula.

Coefficients are the numeric values that multiply the variables in a polynomial. In the quadratic equation written in standard form \( ax^2 + bx + c = 0 \), the coefficients are \(a\), \(b\), and \(c\).

• \(a\) is the coefficient of the \(x^2\) term
• \(b\) is the coefficient of the \(x\) term
• \(c\) is the constant term

For the exercise at hand, after converting the equation into standard form \( x^2 - x + 1 = 0 \), we identified:

• \(a = 1\)
• \(b = -1\)
• \(c = 1\)

Recognizing these coefficients is essential as they are plugged directly into the quadratic formula to find the solutions of the equation.

The discriminant is a part of the quadratic formula that determines the nature of the solutions. It is found inside the square root of the formula:
\( x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \)
The value of the discriminant, \( \Delta \), is given by:
\( \Delta = b^2 - 4ac \)
The discriminant helps us understand what type of solutions we will get:

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), the equation has exactly one real solution.
• If \( \Delta < 0 \), the equation has two complex solutions.

For the exercise, we calculated the discriminant as:
\( \Delta = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \)
Since \( \Delta \) is negative, this indicates that the quadratic equation has two complex solutions.

When dealing with a negative discriminant, the solutions of the quadratic equation are complex numbers. Instead of real solutions, we have solutions involving the imaginary unit \(i\), which is defined as \(i = \sqrt{-1}\).
Given the quadratic formula:
\( x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \), when the discriminant \( \Delta \) is negative, we get imaginary solutions:
In the exercise, the value of the discriminant was found to be \( -3 \). Plugging this into the quadratic formula:
\( x = \frac{-(-1) \, \pm \, \sqrt{-3}}{2 \, (1)} \)
Simplifying further, we get:
\( x = \frac{1 \, \pm \, i \sqrt{3}}{2} \)
Hence, the solutions are:

• \( x = \frac{1 + i \sqrt{3}}{2} \)
• \( x = \frac{1 - i \sqrt{3}}{2} \)

These are the two complex solutions to the quadratic equation.