Lesson Quadratic Equations You Cannot Factor

This Lesson (Quadratic Equations You Cannot Factor) was created by by oberobic(2304)

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Problem: We are given the following equation to solve for x:
.
+x%5E2-9+=+2x+

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Solution: Start by getting the equation into standard form: ax%5E2+%2Bbx+%2B+c+=+0

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In this case, subtract 2x from both sides.
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+x%5E2+-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%28500%2C500%2C-10%2C10%2C-20%2C20%2Cx%5E2-2%2Ax-9%29+

+graph%28500%2C500%2C-10%2C10%2C-20%2C20%2Cx%5E2-2%2Ax-9%29+

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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%5E2+-2x+%28-9%2B9%29+=+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%5E2+-2x+%2B1+=+9+%2B1+

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Now define the squared term to be solved using the half-value of the x-term.
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+%28x-1%29%5E2+=+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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+%28x-1%29+=+sqrt%2810%29+

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Using a calculator you can find the sqrt of 10.
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+sqrt%2810%29+=+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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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation ax%5E2%2Bbx%2Bc=0

(in our case 1x%5E2%2B-2x%2B-9+=+0

) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac

should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac

.

Discriminant d=40 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+40+%29%29%2F2%5Ca

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+40+%29%29%2F2%5C1+=+4.16227766016838

x%5B2%5D+=+%28-%28-2%29-sqrt%28+40+%29%29%2F2%5C1+=+-2.16227766016838

Quadratic expression 1x%5E2%2B-2x%2B-9

can be factored:
1x%5E2%2B-2x%2B-9+=+1%28x-4.16227766016838%29%2A%28x--2.16227766016838%29

Again, the answer is: 4.16227766016838, -2.16227766016838. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-9+%29

.
Answer: We have two roots.
+x+=+-2.16227766016838+

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