Question

What is the solution for the quadratic equation depicted in the graph?
i.imgur.com/gCqO8iE.gif

Answer 1

if you mean the zeroes, then I'd say they are at about (-5,0) and (6,0) making the equation
y = a(x+5)(x-6)
To find a, note that the y-intercept is at about (0,-30) so that means that
-25a = -30
a = 6/5
So, y = 6/5 (x+5)(x-6)

If by "solution" you mean something else, then you need to explain just what you want.

Answer 2

The solution is where y = 0


It is obvious from the figure that one solution is x = 6

I assume (because it is not indicated on the graph) that the second solution is x = - 5 because:

[ - 4 + ( - 6 ) ] / 2 = ( - 4 - 6 ) / 2 = - 10 / 2 = - 5

The roots are x = 6 and x = - 5


Each quadratic equation can be written in the form:

y = a x² + b x + c = a ( x - x1 ) ( x - x2 )

If the solutions are x = 6 and x = - 5 then the quadratic equation is given:

y = a x² + b x + c = a ( x - x1 ) ( x - x2 ) = a ( x - 6 ) [ x - ( - 5 ) ]

y = a ( x - 6 ) ( x + 5 )


The coefficient a can be determined from the conditions x = - 4, y = - 10 ( which can be seen in the diagram ).

put x = - 4 and y = - 10 in the equation:

y = a ( x - 6 ) ( x + 5 )

- 10 = a ( - 4 - 6 ) ( - 4 + 5 )

- 10 = a ∙ ( - 10 ) ∙ 1

- 10 = - 10 a

Divide both sides by - 10

1 = a

a = 1

y = 1 ∙ ( x - 6 ) ( x + 5 )

y = ( x - 6 ) ( x + 5 )

y = x ∙ x - 6 ∙ x + 5 ∙ x - 6 ∙ 5

y = x² - 6 x + 5 x - 30

y = x² - x - 30


y = a x² + b x + c

The coefficients of your equation are:

a = 1 , b = - 1 , c = - 30


The vertex is the lowest point on the parabola if the parabola opens upward and is the highest point on the parabola if the parabola opens downward.

x coordinate of the vertex is:

xv = - b / 2 a

In this case:

xv = - ( - 1 ) / 2 ∙ 1 = 1 / 2

xv = 0.5

yv = ( 1 / 2 )² - 1 / 2 - 30

yv = 1 / 4 - 1 / 2 - 30

yv = - 1 / 4 - 120 / 4

yv = - 121 / 4

yv = - 30.25

Therefore, the vetex is not visible in the graph because the lowest point is drawn on the graph y = - 30


So your function is:

y = x² - x - 30

The roots are ( - 5 , 0 ) , ( 6 , 0 )

The vertex is ( 0.5 , - 30.25 )