## SOLVING QUADRATIC EQUATION BY GRAPHING

Solving quadratic equation by graphing :

Here we are going to see how to how to solve quadratic equation by graphing.

A quadratic function has standard form f(x) = ax2 + bx + c. In a is not equal to zero, the value of the related quadratic function is 0.

## How to Solve quadratic equation by graphing

• To solve quadratic equation by graphing first we need to find the vertex of the given quadratic equation.
• By using the table and giving some random values of x, we can get the values of y.
• Two points of the graph which intersects the x-axis at two distinct points is known as solution.

Note :

Quadratic equations always have two roots. However, these roots are not always two distinct numbers. Sometimes the two roots are the same number.

Let us see some example problems of solving quadratic equation by graphing

## Solving quadratic equation by graphing - Examples

Example 1 :

Solve x2 + 6x + 7 = 0 by graphing

Solution :

Step 1 :

Graph the related function f(x) = x2 + 6x + 7

Let the quadratic function as y = x2 + 6x - 7

x-coordinate of the vertex  =  -b/2a

here a  = 1, b = 6 and c = -7

x  =  -6/2(1) ==> -6/2 ==> -3

Apply x = -3 in the given equation, we get

y = (-3)2 + 6(-3) - 7

y = 9 - 18 - 7

y  =  -16

Hence the vertex is (-3, -16)

Make a table of values to find other points to sketch the graph.

 x-8-6-4-3-202 y9-7-15-16-15-79 Set of ordered pairs :(-8, 9)(-6, -7)(-4, -15)(-3, -16)(-2, -15)(0, -7)(2, 9)

Hence the solution of the quadratic equation are (-7, 1).

Checking :

x2 + 6x - 7 = 0

x2 + 7x - 1x - 7 = 0

x(x + 7) - 1(x + 7) = 0

(x - 1) (x + 7) = 0

x - 1 = 0     x + 7 = 0

x = 1 and x = -7

Let us see next example on"Solving quadratic equation by graphing".

Example 2 :

Solve x2 + x + 4 = 0 by graphing.

Solution :

Step 1 :

Graph the related function f(x) = x2 + x + 4

Let the quadratic function as y = x2 + x + 4

x-coordinate of the vertex  =  -b/2a

here a  = 1, b = 1 and c = 4

x  =  -1/2(1) ==> -1/2

Apply x = -1/2 in the given equation, we get

y = (-1/2)2 + (-1/2) + 4

y = 1/4 - 1/2 + 4

y  =  (1 - 2 + 16)/4

y  =  15/4

Hence the vertex is (-1/2, 15/4)

Make a table of values to find other points to sketch the graph.

 x-1012 y6446 Set of ordered pairs :(-1, 6)(0, 4)(1, 4)(2, 6)

The graph has no x-intercept. Thus, there are no real  number solutions for this equation.

Checking :

x2 + x + 4 = 0

We cannot factor the quadratic equation . Hence it has no real roots.

Let us see next example on"Solving quadratic equation by graphing".

Example 3 :

Solve x2 - 7x + 6 = 0 by graphing.

Solution :

Step 1 :

Graph the related function f(x) = x2 - 7x + 6

Let the quadratic function as y = x2 - 7x + 6

x-coordinate of the vertex  =  -b/2a

here a  = 1, b = -7 and c = 6

x  =  -(-7)/2(1) ==> 7/2 ==> 3.5

Apply x = 7/2 in the given equation, we get

y = (7/2)2 - 7(7/2) + 6

y = 49/4 - 49/2 + 6

y  =  (49 - 98 + 24)/4

y  =  -25/4

Hence the vertex is (7/2, -25/4)

Make a table of values to find other points to sketch the graph.

 x-2-1012 y-41460-4 Set of ordered pairs :(-2, -4)(-1, 14)(0, 6)(1, 0)(2, -4)

Hence the solution of the quadratic equation are (1, 6).

Checking :

x2 - 7x + 6 = 0

x2 - 6x - 1x + 6 = 0

x(x - 6) - 1(x - 6) = 0

(x - 1) (x - 6) = 0

x - 1 = 0     x - 6 = 0

x = 1 and x = 6

After having gone through the stuff given above, we hope that the students would have understood "Solving quadratic equation by graphing".

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