Question

Complete the square to solve 0=x^(2)+9x+16.25 and determine the roots. Then sketch the parabola y=x^(2)+9x+16.25 and state the coordinates of the x intercepts and the vertex.

Answer 1

Answer: x intercepts = -5/2 or x = -13/2

the y intercepts y = 16.5

vertex ( -9/2 , -4)

Explanation:

Answer 2

The roots and x-coordinates of the equation are

`-(5)/(2)` and
`-(13)/(2)`

The vertex of the parabola is

`-4.5`

Explanation:

Given

`0=x^2+9x+16.25\\y=x^2+9x+16.25`

Complete the Square, X intercepts, and Roots

`0=x^2+9x+16.25`

Write
`16.25` as a fraction.

`x^2+9x+16(1)/(4)=0`

`x^2+9x+(65)/(4)=0`

Factor the left hand side.

Write
`x^2` as a fraction with a common denominator, multiply by
`(4)/(4)`.

`x^2*(4)/(4) +9x+(65)/(4)=0`

Combine
`x^2` and
`(4)/(4)`.

`(x^2*4)/(4) +9x+(65)/(4)=0`

Combine the numerators over the common denominator.

`(4x^2+65)/(4) +9x=0`

To write
`9x` as a fraction with a common denominator, multiply by
`(4)/(4)`.

`(4x^2+65)/(4) +9x*(4)/(4) =0`

Combine
`9x` and
`(4)/(4)`.

`(4x^2+65)/(4) +(36x)/(4) =0`

Combine the numerators over the common denominator.

`(36x+4x^2+65)/(4) =0`

For a polynomial of the form
`ax^2+bx+c`, rewrite the middle term as a sum of two terms whose product is
`a*c=4*65=260` and whose sum is
`b=36`.

`(36(x)+4x^2+65)/(4) =0`

Rewrite 36 as 10 plus 26.

`((10+26)x+4x^2+65)/(4) =0`

Apply the distributive property.

`(4x^2+10x+26x+65)/(4) =0`

Group the first two terms and the last two terms.

`(\left(4x^2+10x\right)+26x+65)/(4) =0`

Factor out the GCF from each group.

`(2x\left(2x+5\right)+13\left(2x+5\right))/(4) =0`

Factor the polynomial by factoring out the GCF.

`(\left(2x+5\right)+\left(2x+13\right))/(4) =0`

Set the numerator equal to zero.

If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.

`2x=5=0\\2x=13=0`

Solving for
`x` in each equation gives us

`-(5)/(2)` and
`-(13)/(2)`

The final solution is

`x=-(5)/(2) ,-(13)/(2)`

These values of
`x` are the roots of the equation and lie on the x-axis.

Vertex

We can use the formula
`(-b)/(2a)` to evaluate the vertex.

This formula is for a polynomial of the form
`ax^2+bx+c=0`.

`x^2+9x+16.25=0`

In this case

`a=1\\b=9`

Inserting our values into the equation yields

`(-9)/(2*1)`

`(-9)/(2)=-4.5`