key term - Monic Quadratic Equation

Definition

A monic quadratic equation is a second-degree polynomial equation where the coefficient of the highest degree term, the $x^2$ term, is equal to 1. This type of quadratic equation has the general form $ax^2 + bx + c = 0$, where $a = 1.

5 Must Know Facts For Your Next Test

1. Monic quadratic equations are a special case of quadratic equations where the coefficient of the $x^2$ term is 1.
2. The quadratic formula can be used to solve both monic and non-monic quadratic equations.
3. The discriminant of a monic quadratic equation is still calculated using the formula $b^2 - 4ac$, where $a = 1$.
4. Monic quadratic equations have the same general solution form as non-monic quadratic equations, but the calculations are often simpler due to the coefficient of $x^2$ being 1.
5. Solving monic quadratic equations using the quadratic formula can be more efficient than solving non-monic quadratic equations, as the division by $2a$ is not necessary when $a = 1.

Review Questions

• Explain how the coefficient of the $x^2$ term being 1 simplifies the process of solving a monic quadratic equation using the quadratic formula.
  • When the coefficient of the $x^2$ term, $a$, is equal to 1 in a quadratic equation of the form $ax^2 + bx + c = 0$, the quadratic formula simplifies to $x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$. This is because the division by $2a$ is not necessary when $a = 1$, making the calculation more efficient compared to solving a non-monic quadratic equation.
• Describe how the discriminant of a monic quadratic equation can be used to determine the nature of the solutions.
  • The discriminant of a monic quadratic equation, calculated as $b^2 - 4c$, can be used to determine the nature of the solutions to the equation. If the discriminant is positive, the equation has two real, distinct solutions. If the discriminant is zero, the equation has one real, repeated solution. If the discriminant is negative, the equation has two complex conjugate solutions. This information about the solutions can be crucial when solving and analyzing monic quadratic equations.
• Evaluate how the simplification of the quadratic formula for monic quadratic equations impacts the overall process of solving these types of equations compared to non-monic quadratic equations.
  • The simplification of the quadratic formula for monic quadratic equations, where the coefficient of the $x^2$ term is 1, significantly impacts the overall process of solving these equations compared to non-monic quadratic equations. The absence of the division by $2a$ step makes the calculations more efficient and less prone to errors. Additionally, the simplified formula allows for quicker identification of the nature of the solutions based on the discriminant, as the calculations involved are less complex. This streamlined approach to solving monic quadratic equations can save time and effort, making it a more advantageous method compared to solving non-monic quadratic equations using the full quadratic formula.

Related terms

Quadratic Equation:

A quadratic equation is a polynomial equation of the second degree, in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0.

Quadratic Formula:

The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. The formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Discriminant:

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is the expression $b^2 - 4ac$. The discriminant determines the nature of the solutions to the equation.