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
=
1
Square
This number will make a perfect square on the left side.
Add and subtract 1 to the left side of the equation to keep the balance, then form a square of a binomial
x2+2x-74
=
0
x2+2x+1-1-74
=
0
(x+1)2-1-74
=
0
(x+1)2-114
=
0
Move the constant to the right
(x+1)2-114
=
0
(x+1)2-114+114
=
0+114
Add 114 to both sides
(x+1)2
=
114
Finally, take the square root of both sides and continue solving for x.
Completing the square is done by taking the coefficient of x, 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 to reduce the coefficient of x2
-4x2+21x
=
x+5
-4x2+21x-x
=
x+5-x
Subtract x from both sides
-4x2+20x
=
5
(-4x2+20x)÷(-4)
=
5÷(-4)
Divide both sides by -4
x2-5x
=
-54
Take the coefficient of the middle term, divide it by two and then square it.
x2-5x
=
-54
Coefficient of the middle term
-5÷2
=
-52
Divide it by 2
(-52)2
=
254
Square
This number will make a perfect square on the left side.
Add 254 to both sides to keep the balance, then form a square of a binomial
x2-5x
=
-54
x2-5x+254
=
-54+254
(x-52)2
=
204
(x-52)2
=
5
Finally, take the square root of both sides and continue solving for x.
Completing the square is done by taking the coefficient of x, halving it and then squaring it. Then we add the new value to both sides of the equation.
Simplify the equation, making sure that x2 has 1 as a coefficient
9x2+10x-100
=
4x2+15
9x2+10x-100-4x2
=
4x2+15-4x2
Subtract 4x2 from both sides
5x2+10x-100
=
15
5x2+10x-100+100
=
15+100
Add 100 to both sides
5x2+10x
=
115
5x2+10x÷5
=
115÷5
Divide both sides by 5
x2+2x
=
23
Take the coefficient of the middle term, divide it by two and then square it.
x2+2x
=
23
Coefficient of the middle term
2÷2
=
1
Divide it by 2
(1)2
=
1
Square
This number will make a perfect square on the left side.
Add 1 to both sides to keep the balance, then form a square of a binomial
x2+2x
=
23
x2+2x+1
=
23+1
(x+1)2
=
24
Finally, take the square root of both sides and continue solving for x.
Completing the square is done by taking the coefficient of x, 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 x terms on the right side and factoring it out
y
=
3x2-9x+4
y-4
=
3x2-9x+4-4
Subtract 4 from both sides
y-4
=
3x2-9x
y-4
=
3(x2-3x)
Factor out 3
Take the coefficient of the x term, divide it by two and then square it.
y-4
=
3(x2-3x)
Coefficient of the x term
=
−32
Divide it by 2
(−32)2
=
94
Square
This number will make the x terms a perfect square.
Add and subtract 94 to the grouping of x terms to keep the balance.
y-4
=
3(x2-3x)
y-4
=
3(x2-3x+94-94)
Add and subtract 94
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