Solve the equation by graphing: y=ax square + 3x -2

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nick-teal's profile pic

nick-teal | High School Teacher | (Level 3) Adjunct Educator

Posted on

It does not matter what value we have for a.

We can plug this into the quadratic formula and keep the variable a as part of our answer.

`(-b +- sqrt(b^2 - 4ac))/(2a)`

where a, b and c are 

`ax^2 + bx + c`

in the case of our problem

`y = ax^2 + 3x - 2`

`a = a`

`b = 3` 

`c = -2`

So we plug those values into our quadratic formula

`(-3 +- sqrt(3^2 - 4*a*(-2)))/(2a)`

These are our two solutions.  One plus, one minus.

`x = (-3 + sqrt(9 + 8a))/(2a)`

and

`x = (-3 - sqrt(9 + 8a))/(2a)` 

**IMPORTANT***

If 9+8a is less than zero, the square root will have an imaginary solution, and so there will be no real solutions to this problem.

ishpiro's profile pic

ishpiro | College Teacher | (Level 1) Educator

Posted on

This equation cannot be solved by graphing unless the value of a is known, so the following answer is given assuming the given function is `y = 2x^2 + 3x - 2 ` and the equation needed to be solved is `2x^2 + 3x - 2 = 0 ` .

This is a quadratic function, and its graph is a parabola:

The solutions of the equation are the x-coordinates of the points where the graph intersects x-axis. As can be seen from the graph, these points are (-2,0) and (0.5, 0), so the solutions of the equation are x = -2 and x = 0.5

This can be checked by plugging these values back into the equation and verifying that they yield 0:

` 2*(-2)^2 + 3(-2) - 2 =8 - 6 - 2 = 0 `

`2*(0.5)^2 + 3*0.5 - 2 = 0.5 + 1.5 - 2 = 0 `

So the solutions are x = -2 and x = 0.5.

 

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