Question

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Answer 1

`x = \bold{0.414} \textrm{ and } x= \bold{-2.414}`

Explanation:

See attached graph for a visual

The x-intercepts of a graph are where the graph crosses the x-axis ie where the value of y = 0 The equation of the graph is y=-x^{2}-2x+1

Setting y = 0 and solving for the resultant equation will provide the x-intercepts

`0=-x^(2)-2x+1or-x^(2)-2x+1=0`

Reversing the signs gives us

`x^(2)+2x-1=0`

This is a quadratic equation of the form
`ax^(2)+bx+c=0` with a = 1, b = 2 and c=-1

For an equation of this form the roots of the equation ie the values of x which satisfy the equation are given by

`\frac{-b\pm\sqrt{{b^(2)-4ac}}}{2a}`

Substituting we get the two roots as

`x=\frac{-2\pm\sqrt{{2^(2)-4(1)(-1}}}{2}=\frac{-2\pm\sqrt{{4-(-4)}}}{2}x=(-2\pm√(8))/(2)8 = 4(2)√(8)=√(4)√(2) = 2√(2)`

So the two roots are

`x=(-2\pm2√(2))/(2)`

Dividing by 2 we get the two roots(x-intercepts) as

`x=-1\pm√(2)\\\\x_(1)=-1+√(2) =-1+1.414=\bold{0.414}`

`x_(2)=-1-√(2)=-1-1.414=\boldsymbol{-2.414}`

Answer 2

Answer: About (-2.414, 0) and (0.414, 0)

Explanation:

Let us graph the equation given and see what the x-intercepts are. These are points where the line intercepts the x-axis. See attached.

This means our x-intercepts are at about (-2.414, 0) and (0.414, 0).

These are also equal to
`-1-√(2)` and
`-1+√(2)`