Question

Find the x-intercepts of the graph of the equation. $$y=-2 x^{2}+6 x+9$$

Short answer

The x-intercepts of the graph of the equation will be given by the solutions to this quadratic equation, which are the x-values at which y = 0.

Step-by-step solution

Set the equation equal to zero

First, set the equation equal to zero to solve for x. This will give us the x-values for when y is zero.\[0=-2x^2+6x+9\]

Rearrange the equation

Rearrange this equation to look more like a standard quadratic equation.\[-2x^2 + 6x + 9 = 0\]

Solve the quadratic formula

Now, apply the quadratic formula, which is \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\), where \(a=-2\), \(b=6\), and \(c=9\). This will yield the x-intercepts.

Calculation

Substitute the values of a,b and c into the formula and do the necessary calculations. This will give us two possible values for x, corresponding to the two potential x-intercepts.

Simplify the solutions

Simplify these expressions to obtain the x-coordinates of the x-intercepts on the graph of the equation.