91Ó°ÊÓ

The standard form of a quadratic equation is essential for using the quadratic formula. This form is written as:\[ax^2 + bx + c = 0\]Where:

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

To convert any quadratic equation to this form, move all terms to one side of the equation. For example, starting with:\[ x^2 = 6x - 13 \]We subtract \( 6x \) and add \( 13 \) to both sides, resulting in:\[ x^2 - 6x + 13 = 0 \]Now, it is in the standard form.

The discriminant is a key part of the quadratic formula and plays a crucial role in determining the nature of the solutions. It is given by the expression:\[ b^2 - 4ac \]Here are the possible scenarios based on the value of the discriminant:

• If \( b^2 - 4ac > 0 \), there are two real and distinct solutions.
• If \( b^2 - 4ac = 0 \), there is one real and repeated solution.
• If \( b^2 - 4ac < 0 \), the solutions are complex or imaginary.

For the equation \( x^2 - 6x + 13 = 0 \):\[ b^2 - 4ac = (-6)^2 - 4(1)(13) = 36 - 52 = -16 \]Since the discriminant is \(-16\), we know the solutions will be imaginary.

When the discriminant is negative, the solutions to the quadratic equation are not real numbers; instead, they are imaginary. Imaginary numbers involve the imaginary unit \( i \), where \( i^2 = -1 \).For the quadratic equation \( x^2 - 6x + 13 = 0 \), we calculated the discriminant to be \(-16\). Let’s use this in the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substituting in the values, we get:\[ x = \frac{6 \pm \sqrt{-16}}{2} \]Since \( \sqrt{-16} = 4i \), we have:\[ x = \frac{6 \pm 4i}{2} \]This simplifies to:\[ x = 3 \pm 2i \]Thus, the equation has two imaginary solutions, \( 3 + 2i \) and \( 3 - 2i \). Imaginary solutions always appear in conjugate pairs: \( a + bi \) and \( a - bi \). These solutions extend the concept of solving quadratic equations beyond real numbers, making it possible to solve any quadratic equation.