Solving Quadratic Equations by Graphing

Hind saed, Maryam Abbas, Marwa Nasser 10C

Objectives of the lesson:

• Solve quadratic equations by graphing

• Estimate solutions of quadratic equations by graphing

Quadratic Equations

• Quadratic equations are quadratic functions that re set to equal a value. The standard form of a quadratic equation is , where and and are integers.

• The solutions of a quadratic equation are called the roots of the equation. One method of finding the roots of a quadratic equation is to find the zeros of the related quadratic function.

3 solutions of quadratic equations

The Roots and zeros of the Function

The zeros of the function are x-intercepts of its graph.

Quadratic Function

-2 and 3 are the zeros of the function.

Quadratic Equation

-2 and 3 are the roots of the equation.

(-2,0)

0

f(x)

x

(3,0)

Example 1: Two Real Solutions.

Solve by graphing.Graph the related function, . The equation of the axis of symmetry is or 1.5. Make a table using -values around 1.5. Then graph each point.

The zero of the function is -1 and 4. therefore, the solutions of the equation are -1 and 4 or

x -1 0 1 1.5 2 3 4

f(x) 0 -4 -6 -6.25 -6 -4 0

4

-4

2-2

8

4

-8

x

f(x)

0

Example 2 one real solution

▪ Solve: 14-x²=6x+23▪ 14=x²+6x+23 (add x² to both

sides)▪ 0=x²+6x+9 (subtract 14)▪ Graph the related function

f(x)=x²-6x+9

f(x)

5 4 3 2 1 X

4 1 0 1 4 F(x)

The function only has one zero, 3. Therefore the solution is 3 or {x | x = 3} 0 1 2 43 5

1

3 4

5

Example 3, no real solution

▪ Use a quadratic equation to find two real numbers with a sum of 15 and a product of 63

▪ Let x represent one of the numbers. Then 15-x is the other number

▪ X(15-x)=63▪ 15x-x²=63

▪ -x²+15x-63=0▪ Graph this function▪ The graph has no x-

intercepts. This means the originL equation has no real solution. Thus, it is not possible for two real numbers to have a sum of 15 and a product of 63

f(x)

0 2 4 86 10

8

64

2

10

12

12

10 5 9 6 7.5 X

-13 -13 -9 -9 -6.75 F(x)

14

14

Real-World Example 6: Solve by Using a Calculator.

Relief A package of supplies is tossed form a helicopter at an altitude of 200 feet. The package’s height above the ground is modeled by ,where t is the time in seconds after it is tossed. How long will it take the package to reach the ground?

We need to find t when h(t) is 0. Solve then graph the related function on a graphing calculator.

• Use the zero feature in the CALC menu to find the positive zero in the function, since time cannot be negative

• Use the arrow keys to select a left bound and press Enter.• Locate a right bound and press Enter twice.• The positive zero of the function is about 4.52. The package would take about 4.52

seconds to reach the ground.

The End