Question

1. Complete the square to solve the equation for x: ax^2+bx+c=0. Provide a detailed account of your procedures.

2. Let r and s be numbers such that (x+r)(x+s)=x^2+8x+15.

(a)What must r*s be?

(b)What must r + s be?



3. An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height, s, at time t seconds after launch is, s(t)= -4.9t^2+19.6t+58.8, where s is in meters. How many seconds after launch does the object strike the ground?



4. (Continued) How could you find the vertex of this function without relying on a calculator? What is the object’s maximum height?



5. The graph of the function f(x) = x^2 is translated (shifted) up 3 units and left 4 units. Write an equation to represent the new function after the translations.

Answer 1

Explanation:

1.

`{ax}^(2) + bx + c`

In order to complete the square, we must be in the form

`{x}^(2) + bx + c`

So we divide the first equation by a,

`\frac{ {ax}^(2) + bx + c}{a} = (0)/(a)`

We then get

`{x}^(2) + (b)/(a) x + (c)/(a) = 0`

Next thing, we move the term without a variable( called a constant) to the another side using basic algebra.

`{x}^(2) + (b)/(a) x = - (c)/(a)`

Next, thing we do , we multiply the linear coefficient by 1/2

then square it

`(b)/(a) * (1)/(2) = ((b)/(2a) ) {}^(2) = \frac{b {}^(2) }{4 {a}^(2) }`

We add that term to both sides of the quadratic

`{x}^(2) + (b)/(a) x + \frac{ {b}^(2) }{4 {a}^(2) } = - (c)/(a) + \frac{b {}^(2) }{4 {a}^(2) }`

On the left side, we factor using the Perfect square trinomial.

`(x + (b)/(2a) ) {}^(2) = ( - c)/(a) + \frac{ {b}^(2) }{4 {a}^(2) }`

On the right side, let try to combine these fractions.

Multiply the first fraction by 4a.

We get

`(x + (b)/(2a) ) {}^(2) = \frac{ {b}^(2) - 4ac}{4 {a}^(2) }`

Take the square root of both sides.

`x + (b)/(2a) = \sqrt{ \frac{ {b}^(2) - 4ac}{4 {a}^(2) } }`

Square root of 4a^2. is 2a.

`x + (b)/(2a) = \frac{ \sqrt{ {b}^(2) - 4ac} }{2a}`

`x = - (b)/(2a) ± \frac{ \sqrt{ {b}^(2) - 4ac } }{2a}`

Which is the quadratic formula. Note:Remember that square root have both Positve and negative answers so that why we have ±.