Question

Represent the solution graphically. Check the solution algebraically. $$ -x^{2}+3 x=-4 $$

Short answer

The solutions to the equation are complex numbers \(x=(3±i\sqrt{7})/2\). The graph of the equation is a parabola that opens upwards, without intersection with the x-axis.

Step-by-step solution

Rearrange the Equation

Rearrange the equation into the form of a standard quadratic equation \(ax^{2}+bx+c=0\). Rearranging gives: \(x^{2}-3x+4=0\)

Use the Quadratic Formula

Use the quadratic formula to solve for \(x\). Which gives: \(x=(-(-3)±\sqrt{(-3)^{2}-4*1*4})/2*1\); that yields to \(x= (3±\sqrt{9-16})/2\); thus, \(x=(3±\sqrt{-7})/2\)

Get the Roots

These are complex roots, expressed in terms of 'i', where \(i=\sqrt{-1}\). So, the roots are \(x=(3±i\sqrt{7})/2\)

Check the Solution

Plug each solution into the original equation to confirm they are correct. Given the complexity of the roots, it is expected that both roots will satisfy the equation.

Graphical Representation

The graphical representation of this quadratic equation is a parabola that opens upwards, with a vertex at the point (1.5, -0.25) and intersecting the y-axis at the point (0, 4). However, as the roots of the equation are complex numbers, the graph will not intersect the x-axis