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#### Example

 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.

#### Example

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.

#### Example

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 = x2c 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 = (xc)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.

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