Solving Quadratics – x2 = n, then x = ± n
Up to this point, we have solved quadratic equations using the Zero Product
Property (factoring). We used the following algorithm:
1.
2.
3.
4.
Put everything on one side, zero on the other side
Factor completely
Set each fact equal to zero
Solve the resulting equations
So, if we had a problem like x2 = 16, we used that algorithm.
Given
Step 1
Step 2
Step 3
x2 = 16
x2 – 16 = 0
(x + 4)(x – 4) = 0
x + 4 = 0 or x – 4 = 0
x = – 4 or x = 4
If we did enough of those problems using that algorithm, you might see a
pattern that would allow you to solve the equation in your head by taking the
square root.
Before moving on, let me make a point, this is an equation of degree 2 - a
quadratic equation. That degree suggests that we will have two solutions.
So, if we had the following equations to solve, would you be able to find the
solutions in your head without using the algorithm – without factoring?
1)
x2 = 25
x = ± 25
x= ± 5
2)
x2 = 100
x = ± 100
x = ± 10
3)
x2 = 64
x = ± 64
x= ± 8
My guess is you can solve those in your head. For instance, solving x2 = 36,
most students would not need to put pencil to paper to come up with the
solution
x = ±6
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The very nice thing about math is you can never make it more difficult, only
longer.
Example
Find the solution set for
x2 – 6 = 43
To make this problem look like the ones I just did, let’s isolate the x2.
x2 = 49
x = ±7
To generalize this, let’s construct the following algorithm to solve equations in
the form x2 = n
1.
2.
3.
4.
Example
Find the solution set
Given
Step 1
Step 2
Step 3
Example
Isolate the quadratic term (variable)
Take the square root of both sides,
the right side of the equation is ±
Simplify the radical
Solve the resulting equations
3x2 + 2 = 50
3x2 + 2 = 50
3x2 = 48
x2 = 16
x = ± 16
x= ± 4
Find the solution set
2x2 – 3 = 21
2x2 = 24
x2 = 12
x = ± 12
x = ± 4i3
x = ±2 3
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Example
Find the solution set
(2x + 1)2 + 4 = 29
(2x + 1)2 = 25
2x + 1 = ± 5
2x = ± 5 – 1
2x = +5 – 1 or 2x = –5 – 1
2x = 4
or
2x = – 6
x=2
or
x=–3
Hanlonmath.com
[email protected]
800.218.5482