Real Solutions

Definition

Real solutions refer to the solutions of an equation that have real number values, as opposed to imaginary or complex number solutions. This term is particularly relevant in the context of solving quadratic equations, where the solutions can be real, imaginary, or a combination of both.

5 Must Know Facts For Your Next Test

1. Real solutions to a quadratic equation can be found using the square root property or the quadratic formula.
2. The number and nature of the real solutions to a quadratic equation depend on the values of the coefficients $a$, $b$, and $c$.
3. If the discriminant, $b^2 - 4ac$, is positive, the equation has two real solutions.
4. If the discriminant is zero, the equation has one real solution.
5. If the discriminant is negative, the equation has no real solutions, only imaginary or complex solutions.

Review Questions

• Explain how the square root property can be used to find real solutions to a quadratic equation.
  • The square root property states that if $ax^2 + bx + c = 0$, then $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to find the real solutions to a quadratic equation when the discriminant, $b^2 - 4ac$, is positive. The two real solutions are the positive and negative square roots of the discriminant, divided by $2a$.
• Describe the relationship between the discriminant and the nature of the solutions to a quadratic equation.
  • The discriminant, $b^2 - 4ac$, determines the nature of the solutions to a quadratic equation. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions, only imaginary or complex solutions. This relationship is crucial in understanding when a quadratic equation will have real solutions and when it will not.
• Analyze how the quadratic formula can be used to determine the number and type of solutions to a quadratic equation.
  • The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to find the solutions to a quadratic equation. By examining the discriminant, $b^2 - 4ac$, within the formula, one can determine the number and type of solutions. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions, only imaginary or complex solutions. This analysis of the discriminant is essential in understanding the nature of the solutions to a quadratic equation.

Related terms

Quadratic Equation: A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

Imaginary Solutions: Solutions to an equation that involve the imaginary unit, $i$, where $i^2 = -1$.

Complex Solutions: Solutions to an equation that are a combination of real and imaginary parts, written in the form $a + bi$, where $a$ and $b$ are real numbers.