The table below summarizes the ways you have learned to solve quadratic equations. Although these methods can be used to solve all quadratic equations, each method is most useful in particular situations.
An alternative method is to develop a general formula for
solving any quadratic equation. Begin with the general form of a
quadratic equation, ax2 + bx + c = 0, where a
¹ 0, and complete the square.
Now complete the square.
Finally, take the square root of each side and solve for x.
The result is called the Quadratic Formula and can be used to solve any
quadratic equation.
Use the Quadratic Formula to solve each equation.
1.
x2 – 4x + 3 = 0
Alternative
Solutions:
Check: Substitute values into the original equation.
The
roots are 3 and 1.
2.
–2x2
+ 3x – 5 = 0
Alternative Solutions:
The
square root of a negative number, such as 
, is not a real number. So, there are no real solutions for x.
Check: Use a graphing calculator. It is
clear that the graph of
f(x) = –2x2 + 3x –
5 never crosses the x-axis. Therefore, there
are no real roots for the equation –2x2
+ 3x – 5 = 0.
When the solutions are irrational
numbers, use a calculator to estimate.
Sports
Link
3. “Hang time” is the total amount of time a ball stays in the air. Manuel
can kick a football with an upward velocity of 64 ft/s. His foot meets the ball
2 feet off the ground. What is Manuel’s hang time if the ball is not caught? Use
the formula h(t) = –16t2 + 64t + 2, where h(t)
is the ball’s height in feet for any time t, in seconds, after the ball
is kicked.
Alternative Solutions:
Since time cannot be negative, the only approximate solution
is 4.031. The football has a hang time of about 4 seconds.
Sumber
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