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Here are a few recommended readings before getting started with this lesson.
One afternoon, Jordan was playing soccer in front of her building with some friends. She wanted to invite her neighbor Mark over to play. To do so, she kicked the ball upward as hard as she could, trying to get Mark to see it outside his window. That kick followed the path of the quadratic equation h=6t−t2.
Here, h is the height in meters of the ball above the ground and t is the time in seconds after the kick. If Mark's bedroom window is 10 meters above the ground, could Mark see the ball? If so, find the value of t for the amount of time it takes for the ball to reach Mark's bedroom window. If not, select No real roots.
Substitute values
Calculate power and product
-(-a)=a
Subtract term
Later that evening, she was still in awe at the Quadratic Formula and wrote it on a chalkboard in her room. She then began to think about the cases in which the discriminant is negative. After analyzing the fact that there are two signs in front of the radical term, she came to another brilliant conclusion.
She discovered that if a quadratic equation with real coefficients has one complex solution, the complex conjugate of that solution is another solution as well. In other words, if z=a+bi is a solution to a quadratic equation, then zˉ=a−bi is also a solution. After this, Jordan checked the solutions Mark found and verified that they meet this condition.Substitute values
Calculate power and product
Subtract term
-a=a⋅i
Split into factors
a⋅b=a⋅b
Calculate root
Factor out 2
Simplify quotient
a2+b2=(a+bi)(a−bi)
Maya is reviewing the responses to the last two-question quiz on solving quadratic equations that her math students took through the school's online platform.
Perfect Square Trinomial a2±2ab+b2=(a±b)2 | ||
---|---|---|
Equation | Factorization | Solutions |
x2+6x+9=0 | (x+3)2=0 | x1=x2=-3 |
x2−8x+16=0 | (x−4)2=0 | x1=x2=4 |
x2±2bix−b2=(x±bi)2
Seeing Jordan's enthusiasm for solving quadratic equations, her older sister Dominika wanted to set her an interesting challenge.
Feeling confident, Jordan accepted the challenge.
LHS−4=RHS−4
Split into factors
Rewrite 36 as 62
LHS+25=RHS+25
Rewrite 4 as 22
x2+2bix−b2=(x+bi)2
LHS=RHS
LHS−2i=RHS−2i
Factor each quadratic equation.
In the challenge presented, it was said that Jordan kicked the ball up as hard as she could, trying to make it visible from Mark's bedroom window. Mark's window is 10 meters above the ground. Additionally, it is known that the ball followed the path of the quadratic equation h=6t−t2.
The goal is to determine whether Mark would be able to see the ball. In other words, it should be checked whether the ball can reach a height of at least 10 meters. To find this information, substitute 10 for h into the function rule that describes the path of the ball.Substitute values
Calculate power and product
Subtract term
It can also be determined graphically if the equation 10=6t−t2 has real solutions. To do so, graph y=10 and y=6t−t2 on the same coordinate plane.
As can be seen, the graphs do not intersect, which means that the equation has no real solutions. However, the equation does have imaginary solutions, which cannot be interpreted in the given context. The Quadratic Formula can be applied to find the solutions.LHS+t2−6t=RHS+t2−6t
Use the Quadratic Formula: a=1,b=-6,c=10
Calculate power and product
Subtract term
-(-a)=a
-a=a⋅i
Calculate root
Factor out 2
ba=b