Solve the equation by graphing: y=ax square + 3x -2
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.
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)`
`x = (-3 - sqrt(9 + 8a))/(2a)`
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.