Math problem

Question

Rewrite the quadratic function in standard form by completing the square

 

f(x)= -1/4x^2+5x-2

Answer 1

Dear Kristi,

   Here are the steps:  Y= (-1/4)X2 +5X -2            Remember:  Standard Position:  Y=aX ² +bX +C, where

 

"C" is a constant.

 

1.  Let f(x)=y=0

 

0=(-1/4) X² + 5X -2      Now, add 2 to both sides of the equation.

 

 

 2   =   (-1/4) X² + 5X -2+2      You want a  +1 in the "a" position, so multiply (-1/4) by its reciprocal of (-4/1).  All terms must be multiplied by (-4/1).

 

2*(-4)  = -4(-1/4)X² +  -4 (5)X

 

-8= X² + -20X

 

3.  To complete the square, you must take half the "b" term and square the answer.   (-20/2)²= (-10)² =100.

 

4.  Now add 100 to both sides of the equation:     100-8= X² + -20X + 100 .

 

         92= X² + -20X + 100

 

5.       92= (X-10)²       This is the same as 92= (X- (b/2)).  Take the square root of both sides of the equation to solve for "X."

 

6.     √(92)  = [√(X-10)²]    Note:  The square root of a number squared equals the number itself.  Also, you will get two answers to the square root because a negative times a negative is also a positive.

 

7.  +/- √(92) = X-10        Now add  10 to both sides of equation to solve for "X."

 

   10 +/- √(92)= X    **Here is where you have two solutions for "X."

 

                              10 + √(92)=X          and       10- √(92)=X       You can simplify these answers:

 

                              10+ √(4*23)=X      and          10-  √(4*23)=X

                              10+  2 √(23)=X      and          10- 2√(23)=X                   

 

8.  The final thing that could be done would be to check your answer by substituting the answers one at a time back into the original equation to see if you get y=0.

 

If I have helped you, please give me a thumbs up.  If you have further questions, just make a comment.

                                                            Susan C.

 

   

Answer 2

1. ƒ(x) = -(1/4)x² + 5x -2 ⇐ Original equation.
2. -(1/4)x² + 5x -2 = 0 ⇐ Rewrite to make the equation equal to zero.
3. -(1/4)x² + 5x = 2 ⇐ Move term without any x to the other side by adding or subtracting.
4. x² - 20x = -8 ⇐ Divide both sides of the equation by the term attached to x².
5. x² - 20x + 8 = 0 ⇐ Subtract or add whatever is on the right side of the equation to make the right side equal to zero. Remember to do the exact same thing to the other side of the equation.
6. ƒ(x) = x² - 20x + 8 ⇐ Convert the equation back to its original form. This is the final step if you only wish to rewrite the equation in standard form. If you want to solve the equation, skip steps 5 and 6 (so go back to step 4), then continue the steps below.
7. -20 → -10 ⇐ Take the term attached to x (not x²), then divide it in half and save that for later. Remember its sign!!!!!
8. (-10)² → 100 ⇐ Square the term you saved from step 4.
9. x² - 20x + 100 = 92 ⇐ Add the answer you got from step 5 to both sides of the equation.
10. (x - 10)² = 92 ⇐ Complete the square for the left side of the equation. The number within the parentheses should be the same number you got as an answer in step 7.
11. x - 10 = ±√(92) ⇐ Take the square root of both sides. The right side of the equation could be either positive or negative, so indicate that with a ± sign.
12. x = 10 + √(92) and x = 10 - √(92) ⇐ Solve for x by moving the other term on the left side of the equation to the right by adding or subtracting to both sides. Because you had a ± answer before, this term you moved with be plus or minus that answer.