Completing the square is done by taking the coefficient of xx, halving it and then squaring it. Then we add the new value to both sides of the equation.
Simplify the equation by dividing both sides by 22
2x2-12x2x2−12x
==
-8−8
(2x2-12x)(2x2−12x)÷2÷2
==
-8−8÷2÷2
x2-6xx2−6x
==
-4−4
Take the coefficient of the middle term, divide it by two and then square it.
x2x2-6−6xx
==
-4−4
Coefficient of the middle term
-6−6÷2÷2
==
-3−3
Divide it by 22
(-3)2(−3)2
==
99
Square
This number will make a perfect square on the left side.
Add 99 to both sides of the equation to keep the balance, then form a square of a binomial
x2-6xx2−6x
==
-4−4
x2-6xx2−6x+9+9
==
-4−4+9+9
(x-3)2(x−3)2
==
55
Finally, take the square root of both sides and continue solving for xx.
Completing the square is done by taking the coefficient of xx, halving it and then squaring it. Then we add the new value to both sides of the equation.
Divide the equation by 44 to reduce the coefficient of x2x2
4x2+8x-74x2+8x−7
==
00
4x2+8x-74x2+8x−7÷4÷4
==
00÷4÷4
x2+2x-74x2+2x−74
==
00
Take the coefficient of the middle term, divide it by two and then square it.
x2+x2+22x-74x−74
==
00
Coefficient of the middle term
22÷2÷2
==
11
Divide it by 22
(1)2(1)2
==
11
Square
This number will make a perfect square on the left side.
Add and subtract 11 to the left side of the equation to keep the balance, then form a square of a binomial
x2+2x-74x2+2x−74
==
00
x2+2xx2+2x+1-1+1−1-74−74
==
00
(x+1)2-1-74(x+1)2−1−74
==
00
(x+1)2-114(x+1)2−114
==
00
Move the constant to the right
(x+1)2-114(x+1)2−114
==
00
(x+1)2-114(x+1)2−114+114+114
==
00+114+114
Add 114114 to both sides
(x+1)2(x+1)2
==
114114
Finally, take the square root of both sides and continue solving for xx.
Completing the square is done by taking the coefficient of xx, halving it and then squaring it. Then we add the new value to both sides of the equation.
Simplify the equation and divide the equation by -4−4 to reduce the coefficient of x2x2
-4x2+21x−4x2+21x
==
x+5x+5
-4x2+21x−4x2+21x-x−x
==
x+5x+5-x−x
Subtract xx from both sides
-4x2+20x−4x2+20x
==
55
(-4x2+20x)(−4x2+20x)÷(-4)÷(−4)
==
55÷(-4)÷(−4)
Divide both sides by -4−4
x2-5xx2−5x
==
-54−54
Take the coefficient of the middle term, divide it by two and then square it.
x2x2-5−5xx
==
-54−54
Coefficient of the middle term
-5−5÷2÷2
==
-52−52
Divide it by 22
(-52)2(−52)2
==
254254
Square
This number will make a perfect square on the left side.
Add 254254 to both sides to keep the balance, then form a square of a binomial
x2-5xx2−5x
==
-54−54
x2-5xx2−5x+254+254
==
-54−54+254+254
(x-52)2(x−52)2
==
204204
(x-52)2(x−52)2
==
55
Finally, take the square root of both sides and continue solving for xx.
Completing the square is done by taking the coefficient of xx, halving it and then squaring it. Then we add the new value to both sides of the equation.
Simplify the equation, making sure that x2x2 has 11 as a coefficient
9x2+10x-1009x2+10x−100
==
4x2+154x2+15
9x2+10x-1009x2+10x−100-4x2−4x2
==
4x2+154x2+15-4x2−4x2
Subtract 4x24x2 from both sides
5x2+10x-1005x2+10x−100
==
1515
5x2+10x-1005x2+10x−100+100+100
==
1515+100+100
Add 100100 to both sides
5x2+10x5x2+10x
==
115115
5x2+10x5x2+10x÷5÷5
==
115115÷5÷5
Divide both sides by 55
x2+2xx2+2x
==
2323
Take the coefficient of the middle term, divide it by two and then square it.
x2+x2+22xx
==
2323
Coefficient of the middle term
22÷2÷2
==
11
Divide it by 22
(1)2(1)2
==
11
Square
This number will make a perfect square on the left side.
Add 11 to both sides to keep the balance, then form a square of a binomial
x2+2xx2+2x
==
2323
x2+2xx2+2x+1+1
==
2323+1+1
(x+1)2(x+1)2
==
2424
Finally, take the square root of both sides and continue solving for xx.
Completing the square is done by taking the coefficient of xx, halving it and then squaring it. Then we add the new value to both sides of the equation.
To find the vertex, transform the given function into vertex form
Start by leaving xx terms on the right side and factoring it out
yy
==
3x2-9x+43x2−9x+4
yy-4−4
==
3x2-9x+43x2−9x+4-4−4
Subtract 44 from both sides
y-4y−4
==
3x2-9x3x2−9x
y-4y−4
==
3(x2-3x)3(x2−3x)
Factor out 33
Take the coefficient of the xx term, divide it by two and then square it.
y-4y−4
==
3(x23(x2-3−3x)x)
Coefficient of the xx term
==
−32−32
Divide it by 22
(−32)2(−32)2
==
9494
Square
This number will make the xx terms a perfect square.
Add and subtract 9494 to the grouping of xx terms to keep the balance.
y-4y−4
==
3(x2-3x)3(x2−3x)
y-4y−4
==
3(x2-3x+3(x2−3x+9494-−9494))
Add and subtract 9494
Now, transform the grouping of x terms into a square of a binomial.
[show cross method with two x’s on the left and two -32s on the right]
y-4
=
3[(x-32)(x-32)-94]
y-4
=
3[(x-32)2-94]
Now, distribute 3 and leave y on the left side
y-4
=
3[(x-32)2-94]
y-4
=
3(x-32)2-3(94)
Distribute 3
y-4
=
3(x-32)2-274
y-4+4
=
3(x-32)2-274+4
Add 4 to both sides
y
=
3(x-32)2-114
Finally, the function is in vertex form
Compare the function to the general vertex form to get the vertex