Quadratic Trinomials, Quadratic Equations, and the Quadratic Formula Help (page 2)
Introduction to Qudratic Trinomials and Equations
A quadratic trinomial contains an x2 term as well as an x term. For example, x2 – 6x + 8 is a quadratic trinomial. You can factor quadratic trinomials by using the FOIL method in reverse.
Let's factor x2 – 6x + 8.
Start by looking at the last term in the trinomial: 8. Ask yourself, "What two integers, when multiplied together, have a product of positive 8?" Make a mental list of these integers:
1 ×8 –1× –8 2 ×4 –2× –4
Next look at the middle term of the trinomial: –6x. Choose the two factors from the list you just made that also add up to the coefficient –6:
–2 and –4
Now write the factors using –2 and –4:
(x – 2)(x – 4)
Use the FOIL method to double-check your answer:
(x – 2)(x – 4) = x2 – 6x + 8
You can see that the answer is correct.
A quadratic equation is an equation that does not graph into a straight line. The graph will be a smooth curve. An equation is a quadratic equation if the highest exponent of the variable is 2. Here are some examples of quadratic equations:
x2 + 6x + 10 = 0
6x2 + 8x – 22 = 0
A quadratic equation can be written in the form: ax2+ bx + c = 0. The a represents the number in front of the x2 variable. The b represents the number in front of the x variable and c is the number. For instance, in the equation 2x2 + 3x + 5 = 0, the a is 2, the b is 3, and the c is 5. In the equation 4x2 – 6x + 7 = 0, the a is 4, the b is –6, and the c is 7. In the equation 5x2 + 7 = 0, the a is 5, the b is 0, and the c is 7. In the equation 8x2 – 3x = 0, the a is 8, the b is –3, and the c is 0. Is the equation 2x + 7 = 0 a quadratic equation? No! The equation does not contain a variable with an exponent of 2. Therefore, it is not a quadratic equation.
Solving Quadratic Equations Using Factoring
Why is the equation x2 = 4 a quadratic equation? It is a quadratic equation because the variable has an exponent of 2. To solve a quadratic equation, first make one side of the equation zero. Let's work with x2 = 4.
Subtract 4 from both sides of the equation to make one side of the equation zero: x2 – 4 = 4 – 4. Now, simplify x2 – 4 = 0. The next step is to factor x2 – 4. It can be factored as the difference of two squares: (x – 2)(x + 2) = 0.
If ab = 0, you know that either a or b or both factors have to be zero because a times b = 0. This is called the zero product property, and it says that if the product of two numbers is zero, then one or both of the numbers have to be zero. You can use this idea to help solve quadratic equations with the factoring method.
Use the zero product property, and set each factor equal to zero: (x – 2) = 0 and (x + 2) = 0.
When you use the zero product property, you get linear equations that you already know how to solve.
Solve the equation: x – 2 = 0 Add 2 to both sides of the equation. x – 2 + 2 = 0 + 2 Now, simplify: x = 2 Solve the equation: x + 2 = 0 Subtract 2 from both sides of the equation. x + 2 – 2 = 0 – 2 Simplify: x = –2
You got two values for x. The two solutions for x are 2 and –2. All quadratic equations have two solutions. The exponent 2 in the equation tells you that the equation is quadratic, and it also tells you that you will have two answers.
Tip: When both your solutions are the same number, this is called a double root. You will get a double root when both factors are the same.
Before you can factor an expression, the expression must be arranged in descending order. An expression is in descending order when you start with the largest exponent and descend to the smallest, as shown in this example: 2x2 + 5x + 6 = 0.
All quadratic equations have two solutions. The exponent of 2 in the equation tells you to expect two answers.
|x2 – 3x – 4 = 0|
|Factor the trinomial x2 –3x – 4.||(x – 4)(x + 1) = 0|
|Set each factor equal to zero.||x – 4=0 and x + 1= 0|
|Solve the equation.||x – 4 = 0|
|Add 4 to both sides of the equation.||x – 4 + 4 = 0 + 4|
|Simplify.||x = 4|
|Solve the equation.||x + 1 = 0|
|Subtract 1 from both sides of the equation.||x + 1 – 1 = 0 – 1|
|Simplify.||x = –1|
|The two solutions for the quadratic equation are 4 and –1.|
Tip: When you have an equation in factor form, disregard any factor that is a number and contains no variables. For example, in 4(x – 5) (x + 5) = 0, disregard the 4. It will have no effect on your two solutions.
Solving Quadratic Equations by Using the Zero Product Rule
If a quadratic equation is not equal to zero, rewrite it so that you can solve it using the zero product rule.
If you need to solve x2 – 11x = 12, subtract 12 from both sides:
x2 – 11x – 12 = 12 – 12
x2 – 11x – 12 = 0
Now this quadratic equation can be solved using the zero product rule.
A quadratic equation must be factored before using the zero product rule to solve it.
To solve x2 + 9x = 0, first factor it:
x(x + 9) = 0
Now you can solve it.
Either x = 0 or x + 9 = 0.
Therefore, possible solutions are x = 0 and x = –9.
Graphs of Quadratic Equations - Parabolas
Introduction to Parabolas
The (x,y) solutions to quadratic equations can be plotted on a graph. These graphs are called parabolas. Typically you will be presented with parabolas given by equations in the form of y = ax2 + bx + c.
Notice that the equation y = x2 conforms to this formula—both b and c are zero.
y = (1)x2 + (0)x + (0) is equivalent to y = x2
The value of a cannot equal zero, however.
Movement of the Parabola on the Graph - Opening Up and Down
If a is greater than zero, the parabola will open upward. If a is less than zero, the parabola will open downward.
The x-coordinate of the turning point, or vertex, of the parabola is given by:
You can use this x-value in the original formula and solve for y (the y-coordinate of the turning point).
There will also be a line of symmetry given by:
For the graph y = x2, x = = 0. The line of symmetry is x = 0. The y-coordinate of the vertex is at located at y = x2 = 02 = 0, so the vertex is at (0,0). Technically a parabola could also be given by the formula x = ay2 + by + c.
The graph of the equation y = x2 is a parabola.
Because the x-value is squared, the positive values of x yield the same y-values as the negative values of x. The graph of y = x2 has its vertex at the point (0,0). The vertex of a parabola is the turning point of the parabola. It is either the minimum or maximum y-value of the graph. The graph of y = x2 has its minimum at (0,0). There are no y-values less than 0 on the graph.
Movement of the Parabola on the Graph - Moving Up, Down, Left, and Right
The graph of y = x2 can be translated around the coordinate plane. While the parabola y = x2 has its vertex at (0,0), the parabola y = x2 – 1 has its vertex at (0,–1). After the x term is squared, the graph is shifted down one unit. A parabola of the form y = x2 – c has its vertex at (0,–c) and a parabola of the form y = x2 + c has its vertex at (0,c).
The parabola y = (x + 1)2 has its vertex at (–1,0). The x-value is increased before it is squared. The minimum value of the parabola is when y = 0 (because y = (x + 1)2 can never have a negative value). The expression (x + 1)2 is equal to 0 when x = –1. A parabola of the form y = (x – c)2 has its vertex at (c,0) and a parabola of the form y = (x + c)2 has its vertex at (0,–c).
What are the coordinates of the vertex of the parabola formed by the equation y = (x – 2)2 + 3?
To find the x-value of the vertex, set (x – 2) equal to 0: x – 2 = 0, x = 2. The y-value of the vertex of the parabola is equal to the constant that is added to or subtracted from the x squared term. The y-value of the vertex is 3, making the coordinates of the vertex of the parabola (2,3).
If parabolas with the formula y = x2 + bx + c open upward or downward, how do you think parabolas given by the formula x = ay2 + by + c appear?
It is important to be able to look at an equation and understand what its graph will look like. You must be able to determine what calculation to perform on each x-value to produce its corresponding y-value.
For example, here is the graph of y = x2.
The equation y = x2 tells you that for every x-value, you must square the x-value to find its corresponding y-value. Let's explore the graph with a few x-coordinates:
An x-value of 1 produces what y-value? Plug x = 1 into y = x2. When x = 1, y = 12, so y = 1. So, you know a coordinate in the graph of y = x2 is (1,1).
An x-value of 2 produces what y-value? Plug x = 2 into y = x2. When x = 2, y = 22, so y = 4. Therefore, you know a coordinate in the graph of y = x2 is (2,4).
An x-value of 3 produces what y-value? Plug x = 3 into y = x2. When x = 3, y = 32, so y = 9. That determines that a coordinate in the graph of y = x2 is (3,9).
Tip: Solving the formula of a parabola for x tells you the x intercept (or intercepts) of the parabola—that is, where the parabola crosses the x-axis. If you get two real values for x, the parabola crosses the x-axis at two points. If you get one real root, then that value is the vertex. If both roots are complex, then the parabola never crosses the x-axis.