# the quadratic formula

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```					Lesson 7.4
   Although you can always use a graph of a
quadratic function to approximate the x-
intercepts, you are often not able to find
exact solutions. This lesson will develop a
procedure to find the exact roots of a
quadratic equation. Consider again this
situation from Example C in the previous
lesson.
   Nora hits a softball straight up at a speed of 120
ft/s. Her bat contacts the ball at a height of 3 ft
above the ground. Recall that the equation relating
height in feet, y, and time in seconds, x, is
y=-16x2 +120x+3. How long will it be until the
ball hits the ground?
   The height will be zero when the ball hits the
ground, so you want to find the solutions to the
equation -16x2 +120x+3=0. You can approximate
the x-intercepts by graphing, but you may not be
able to find the exact x-intercept.
   You will not be able to factor this equation using a rectangle
diagram, so you can’t use the zero-product property.
   Instead, to solve this equation symbolically, first write the
equation in the form a(x-h)2+k=0.

The zeros of the
function are
x=7.525 and
x=- 0.025.
The negative time,
-0.025 s, does not
make sense in this
situation, so the ball
hits the ground after
approximately
7.525 s.
   If you follow the
same steps with a
general quadratic
equation, then you
can develop the
quadratic formula.
   This formula provides
solutions to ax2 +bx + c
=0 in terms of a, b, and
c.
   To use the quadratic formula on the equation
in Example A, 16x2+120x +3= 0, first
identify the coefficients as a=16, b=120, and
c=3. The solutions are
   Salvador hits a baseball at a height of 3 ft and
with an initial upward velocity of 88 feet per
second.
◦ Let x represent time in seconds after the ball is hit,
and let y represent the height of the ball in feet.
Write an equation that gives the height as a
function of time.
◦ Y=-16x2+88x+3
   Write an equation to find the times when the
ball is 24 ft above the ground.
◦ 24=-16x2+88x +3
   Rewrite your equation from the previous
step in the form, ax2 +bx + c =0 then use
the quadratic formula to solve. What is the
real-world meaning of each of your
solutions? Why are there two solutions?
◦ -16x2+88x -21=0
◦ X=0.25 or x=5.25.
◦ The ball rises to 24 ft on the way up at 0.25 s, and it falls
to 24 ft on the way down at 5.25 s.
 Find the y-coordinate of the vertex of this
parabola.
 How many different x-values correspond to
this y-value? Explain.
◦ 124
◦ The ball reaches the maximum height only once.
The ball reaches other heights once on the way up
and once on the way down, but the top of the ball’s
height, the maximum point, can be reached only
once.
   Write an equation to find the time when the ball
reaches its maximum height.
◦ Use the quadratic formula to solve the equation.
At what point in the solution process does it
become obvious that there is only one solution to
this equation?
-16x2 +88x +3=124; x=2.75; when b2  4ac  0
   Write an equation to find the time when the ball
reaches a height of 200 ft. What happens when
you try to solve this impossible situation with the
quadratic formula?
-16x2 +88x +3= 200. You get the square root of a
negative number, so there are no real solutions.
   Solve 3x2=5x+8.
   To use the quadratic formula, first write the equation in the
form ax2+bx+c=0 and identify the coefficients: 3x2-5x-8=0
   a=3, b=5, c=8
   Substitute a, b, and c into the quadratic formula.
   The solutions are
x=8/3 or x=-1.
Remember, you can find exact solutions to
some quadratic equations by factoring.
However, most quadratic equations don’t
factor easily. The quadratic formula can be
used to solve any quadratic equation.

```
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