SOLUTION: Solve by completing the square. 4x^2 + 2x – 3 = 0

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Question 84864: Solve by completing the square.
4x^2 + 2x – 3 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=4+x%5E2%2B2+x-3 Start with the given equation



y%2B3=4+x%5E2%2B2+x Add 3 to both sides



y%2B3=4%28x%5E2%2B%281%2F2%29x%29 Factor out the leading coefficient 4



Take half of the x coefficient 1%2F2 to get 1%2F4 (ie %281%2F2%29%281%2F2%29=1%2F4).


Now square 1%2F4 to get 1%2F16 (ie %281%2F4%29%5E2=%281%2F4%29%281%2F4%29=1%2F16)





y%2B3=4%28x%5E2%2B%281%2F2%29x%2B1%2F16-1%2F16%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1%2F16 does not change the equation




y%2B3=4%28%28x%2B1%2F4%29%5E2-1%2F16%29 Now factor x%5E2%2B%281%2F2%29x%2B1%2F16 to get %28x%2B1%2F4%29%5E2



y%2B3=4%28x%2B1%2F4%29%5E2-4%281%2F16%29 Distribute



y%2B3=4%28x%2B1%2F4%29%5E2-1%2F4 Multiply



y=4%28x%2B1%2F4%29%5E2-1%2F4-3 Now add %2B3 to both sides to isolate y



y=4%28x%2B1%2F4%29%5E2-13%2F4 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=4, h=-1%2F4, and k=-13%2F4. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=4x%5E2%2B2x-3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B2x-3%29 Graph of y=4x%5E2%2B2x-3. Notice how the vertex is (-1%2F4,-13%2F4).



Notice if we graph the final equation y=4%28x%2B1%2F4%29%5E2-13%2F4 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4%28x%2B1%2F4%29%5E2-13%2F4%29 Graph of y=4%28x%2B1%2F4%29%5E2-13%2F4. Notice how the vertex is also (-1%2F4,-13%2F4).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.






Since we have

y=4%28x%2B1%2F4%29%5E2-13%2F4

Set y=0 and solve for x
0=4%28x%2B1%2F4%29%5E2-13%2F4

13%2F4=4%28x%2B1%2F4%29%5E2 Add 13%2F4 to both sides


%2813%2F4%29%281%2F4%29=%28x%2B1%2F4%29%5E2 Multiply both sides by 1%2F4

0%2B-sqrt%2813%2F16%29=x%2B1%2F4 Take the square root of both sides


0%2B-sqrt%2813%29%2F4-1%2F4=x Subtract 1%2F4 from both sides

So our answer is
x=%28-1%2B-sqrt%2813%29%29%2F4
which breaks down to
x=%28-1%2Bsqrt%2813%29%29%2F4 or x=%28-1-sqrt%2813%29%29%2F4