Lesson Quadratic Equations You Cannot Factor

Source code of 'Quadratic Equations You Cannot Factor'

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Problem: We are given the following equation to solve for x:
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{{{ x^2-9 = 2x }}}
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Solution: Start by getting the equation into standard form: {{{ax^2 +bx + c = 0}}}
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In this case, subtract 2x from both sides.
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{{{ x^2 -2x -9 = 2x-2x = 0 }}}
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We can look for integer multiplicative factors of -9 that would leave us with -2 when added or subtracted.
9 = 1*9
9-1 = 8
-9+1 = -8
9 = 3*3
9-3 = 6
-9+3=-6
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So, we cannot simply factor this equation to solve it.
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At this point, looking at the graph is helpful.
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{{{ graph(500,500,-10,10,-20,20,x^2-2*x-9) }}}
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We can see there are two roots (zeroes) of x.
So, our choices are to complete the square or to use the quadratic equation.
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Completing the Square
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Isolate the 'c-term' on the right-hand side of the equation.
To do this, add 9 to both sides.
{{{ x^2 -2x (-9+9) = 9}}}
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{{{ x^2 -2x = 9}}}
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Divide the x-term's coefficient ('b') in half.
-2/2 = -1
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Square this number and add to both sides of the equation
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{{{ x^2 -2x +1 = 9 +1 }}}
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Now define the squared term to be solved using the half-value of the x-term.
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{{{ (x-1)^2 = 10 }}}
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Since the left-hand side is squared now, the solution is the square root of the right-hand side.
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{{{ (x-1) = sqrt(10) }}}
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Using a calculator you can find the sqrt of 10.
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{{{ sqrt(10) = 3.16227766016838 }}}
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Remember the sqrt term has both plus or minus values.
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Substitute the plus or minus values and solve to find the two zeroes (roots) for 'x'.
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{{{ x -1 = -3.16227766016838 }}}
Add 1 to both sides.
{{{ x = -2.16227766016838 }}}
This defines the point: (-2.16227766016838, 0).
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{{{ x-1 = 3.16227766016838 }}}
Add 1 to both sides.
{{{ x = 4.16227766016838 }}}
This defines the point: (4.16227766016838, 0).
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Looking back at the graph, you can see the parabola crosses the x-axis at these points.
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Quadratic Equation
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As another approach, you could use the quadratic equation.
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*[invoke quadratic "x", 1, -2, -9 ]
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Answer: We have two roots.
{{{ x = -2.16227766016838 }}}
{{{ x = 4.16227766016838 }}}