SOLUTION: 5x^2+9x-2=0 write the related function in vertex form, use the quadratic formula to find the roots and graph the parbola

Algebra ->  Trigonometry-basics -> SOLUTION: 5x^2+9x-2=0 write the related function in vertex form, use the quadratic formula to find the roots and graph the parbola      Log On


   

Question 124683: 5x^2+9x-2=0 write the related function in vertex form, use the quadratic formula to find the roots and graph the parbola
Answer by MathLover1(20850) 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=5+x%5E2%2B9+x-2 Start with the given equation



y%2B2=5+x%5E2%2B9+x Add 2 to both sides



y%2B2=5%28x%5E2%2B%289%2F5%29x%29 Factor out the leading coefficient 5



Take half of the x coefficient 9%2F5 to get 9%2F10 (ie %281%2F2%29%289%2F5%29=9%2F10).


Now square 9%2F10 to get 81%2F100 (ie %289%2F10%29%5E2=%289%2F10%29%289%2F10%29=81%2F100)





y%2B2=5%28x%5E2%2B%289%2F5%29x%2B81%2F100-81%2F100%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 81%2F100 does not change the equation




y%2B2=5%28%28x%2B9%2F10%29%5E2-81%2F100%29 Now factor x%5E2%2B%289%2F5%29x%2B81%2F100 to get %28x%2B9%2F10%29%5E2



y%2B2=5%28x%2B9%2F10%29%5E2-5%2881%2F100%29 Distribute



y%2B2=5%28x%2B9%2F10%29%5E2-81%2F20 Multiply



y=5%28x%2B9%2F10%29%5E2-81%2F20-2 Now add %2B2 to both sides to isolate y



y=5%28x%2B9%2F10%29%5E2-121%2F20 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=5, h=-9%2F10, and k=-121%2F20. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=5x%5E2%2B9x-2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C5x%5E2%2B9x-2%29 Graph of y=5x%5E2%2B9x-2. Notice how the vertex is (-9%2F10,-121%2F20).



Notice if we graph the final equation y=5%28x%2B9%2F10%29%5E2-121%2F20 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C5%28x%2B9%2F10%29%5E2-121%2F20%29 Graph of y=5%28x%2B9%2F10%29%5E2-121%2F20. Notice how the vertex is also (-9%2F10,-121%2F20).



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