Glossary
Completing the Square
A method for solving quadratic equations by manipulating the equation to create a perfect square trinomial on one side. This allows for taking the square root of both sides to find the solutions.
Example:
To solve x² - 10x + 21 = 0, you could use Completing the Square by adding 25 to both sides to make the left side (x-5)², then solving for x.
Discriminant
The expression b² - 4ac found under the square root in the quadratic formula. Its value determines the number and type of real solutions a quadratic equation has.
Example:
When analyzing x² + 2x + 3 = 0, calculating the Discriminant (2² - 413 = 4 - 12 = -8) tells us there are no real roots.
Factoring
A method for solving quadratic equations by breaking down the quadratic expression into a product of two linear factors. This method relies on the Zero Product Property.
Example:
To solve x² + 7x + 10 = 0, you can use factoring to get (x+2)(x+5) = 0, leading to solutions x = -2 and x = -5.
Factoring with a = 1
A specific factoring technique for quadratic equations in standard form x² + bx + c = 0, where the leading coefficient 'a' is 1. You look for two numbers that multiply to 'c' and add to 'b'.
Example:
For x² + 8x + 15 = 0, Factoring with a = 1 involves finding two numbers that multiply to 15 and add to 8, which are 3 and 5, leading to (x+3)(x+5)=0.
Factoring with a ≠ 1
A factoring technique for quadratic equations where the leading coefficient 'a' is not 1. This often involves methods like grouping or the 'ac' method, sometimes after factoring out a GCF.
Example:
To factor 2x² + 11x + 5 = 0, which is an example of Factoring with a ≠ 1, you might look for two numbers that multiply to (2*5)=10 and add to 11, then use grouping.
Leading Coefficient
The coefficient of the term with the highest degree in a polynomial, specifically 'a' in the standard form of a quadratic equation ax² + bx + c = 0.
Example:
In the equation 5x² - 3x + 7 = 0, the Leading Coefficient is 5.
Negative Discriminant
Occurs when the value of b² - 4ac is less than zero. This indicates that the quadratic equation has no real roots, but instead has two complex (imaginary) roots.
Example:
If you calculate the discriminant for x² + x + 1 = 0 and get -3, this Negative Discriminant tells you there are no real solutions.
Perfect Square Trinomial
A trinomial that results from squaring a binomial, such as (x + k)² or (x - k)². It has the form x² + 2kx + k² or x² - 2kx + k².
Example:
The expression x² + 6x + 9 is a Perfect Square Trinomial because it can be factored as (x + 3)².
Positive Discriminant
Occurs when the value of b² - 4ac is greater than zero. This indicates that the quadratic equation has two distinct real roots.
Example:
For x² + 5x + 6 = 0, the discriminant is 5² - 4(1)(6) = 25 - 24 = 1. Since 1 is a Positive Discriminant, there are two distinct real roots.
Quadratic Equation
An equation of the second degree, meaning it contains at least one term in which the unknown variable is squared. It is typically written in the standard form ax² + bx + c = 0.
Example:
When a ball is thrown, its height over time can be modeled by a quadratic equation like h(t) = -16t² + 64t + 5.
Quadratic Formula
A universal formula used to find the solutions (roots) of any quadratic equation in standard form ax² + bx + c = 0. The formula is x = [-b ± √(b² - 4ac)] / 2a.
Example:
If you can't easily factor 3x² - 2x - 8 = 0, the Quadratic Formula will always provide the correct solutions.
Roots (or Solutions)
The values of the variable that satisfy a quadratic equation, meaning they make the equation true. Graphically, these are the x-intercepts where the parabola crosses the x-axis.
Example:
For the equation x² - 9 = 0, the Roots are x = 3 and x = -3, as these are the values that make the equation true.
Standard Form
The conventional way to write a quadratic equation, expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' are real numbers and 'a' is not equal to zero.
Example:
Before applying the quadratic formula, ensure your equation, like 2x² = 5x - 3, is rearranged into Standard Form: 2x² - 5x + 3 = 0.
Zero Discriminant
Occurs when the value of b² - 4ac is exactly zero. This indicates that the quadratic equation has exactly one real root, which is a double root.
Example:
The equation x² - 4x + 4 = 0 has a Zero Discriminant ((-4)² - 4(1)(4) = 16 - 16 = 0), meaning it has only one real root (x=2).
Zero Product Property
A fundamental principle stating that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is crucial for solving equations by factoring.
Example:
Using the Zero Product Property, if (x-4)(x+1) = 0, then either x-4 = 0 or x+1 = 0, meaning x = 4 or x = -1.