Quadratic equations

Definition

A quadratic equation is a second-degree polynomial equation that can be expressed in the standard form $$ax^2 + bx + c = 0$$, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. Quadratic equations can represent various phenomena in mathematics and the real world, and their solutions, known as roots, can be found using methods such as factoring, completing the square, or applying the quadratic formula. The graphical representation of a quadratic equation is a parabola, which opens either upwards or downwards depending on the value of 'a'.

5 Must Know Facts For Your Next Test

1. Quadratic equations can have two, one, or no real roots depending on the value of the discriminant.
2. The quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ is used to find the roots of any quadratic equation.
3. Factoring a quadratic equation can simplify finding its roots when it can be expressed as a product of two binomials.
4. Completing the square is another method used to derive the quadratic formula and solve for the roots of a quadratic equation.
5. The vertex of the parabola represented by a quadratic equation gives important information about its maximum or minimum value.

Review Questions

• How do you determine the number and type of roots of a quadratic equation using the discriminant?
  • The discriminant is calculated using the formula $$b^2 - 4ac$$ from the standard form of a quadratic equation. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (the vertex touches the x-axis). If the discriminant is negative, there are no real roots; instead, there are two complex roots. This provides insight into the behavior of the parabola in relation to the x-axis.
• Discuss how completing the square can be used to derive the quadratic formula and solve for roots.
  • Completing the square involves rearranging a quadratic equation into a perfect square trinomial. Starting with $$ax^2 + bx + c = 0$$, you divide through by 'a' and manipulate it to isolate constants. By adding and subtracting the square of half of 'b/a', you transform it into $$a(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a}$$. From here, taking square roots on both sides and isolating 'x' leads directly to the quadratic formula, enabling efficient root calculation.
• Evaluate how different values of 'a' in a quadratic equation affect its graph and solutions.
  • The coefficient 'a' in a quadratic equation determines both the direction and width of its parabola. If 'a' is positive, the parabola opens upwards, indicating that it has a minimum point (the vertex). Conversely, if 'a' is negative, it opens downwards, showcasing a maximum point. Furthermore, larger absolute values of 'a' result in narrower parabolas while smaller absolute values create wider ones. This directly influences how many times the parabola intersects the x-axis and thus affects the number of real solutions to the equation.

Related terms

Roots: The values of 'x' that satisfy the quadratic equation, representing the points where the parabola intersects the x-axis.

Discriminant: A part of the quadratic formula given by $$b^2 - 4ac$$ that helps determine the nature of the roots (real or complex) of a quadratic equation.

Parabola: The U-shaped graph of a quadratic function, which can either open upwards if 'a' is positive or downwards if 'a' is negative.