Solving Quadratic Equations

by Graphing

Quadratic Equation

y = ax2 + bx + c

• ax2 is the quadratic term.• bx is the linear term.• c is the constant term.

The highest exponent is two; therefore, the degree is two.

Solving Equations

When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts.

These values are also referred to as solutions, zeros, or roots.

The number of real solutions is at most two.

Quadratic Solutions

No solutions

6

4

2

-2

5

f x = x2-2 x +5

6

4

2

-2

5

2

-2

-4

-5 5

One solution Two solutions

Example f(x) = x2 - 4

Identifying Solutions4

2

-2

-4

-5 5

Solutions are -2 and 2.

Example f(x) = 2x -

x2

Solutions are 0 and 2.

Identifying Solutions

4

2

-2

-4

5

VertexWhen we are looking at real life examples like the motion of a ball thrown in the air, you can use the graph of the quadratic equation to gain information. We usually only focus on the 1st quadrant. y

x

VertexThe max height of the ball is 2 ft, and it reaches that height after 1 sec.

The ball leaves the ground at 0 sec and returns to the ground after 2 sec.

Height

in ft

Time

in sec

Graphing a Quadratic Function

Remember, the steps to graphing a parabola in standard form:

STEP 1: Find the Axis of symmetry using:

STEP 2: Find the vertex

STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.

MAKE A TABLE

using x – values close to the Axis of symmetry.

2ba

x

The graph of a quadratic equation is a parabola.

The roots or zeros are the x-intercepts.

The vertex is the maximum or minimum point.

All parabolas have an axis of symmetry.

Graphing Quadratic Equations

Another method of graphing uses a table

with

arbitrary x-values.Graph y = x2 - 4x

Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2

Graphing Quadratic Equations

x y0 01 -32 -43 -34 0

4

2

-2

-4

5