Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.
Plot the vertex and x intercepts, then connect the points to form a parabola
First, find the vertex and plot this on the graph
Start by transforming the function into vertex form
y
=
x2-10x+15
y-15
=
x2-10x+15-15
Subtract 15 from both sides
y-15
=
x2-10x
Take the coefficient of the x term, divide it by two and then square it.
y-15
=
x2-10x
Coefficient of the x term
=
โ102
Divide it by 2
(โ5)2
=
25
Square
This number will make the right side a perfect square.
Add and subtract 25 to the right side to keep the balance.
y-15
=
x2-10x
y-15
=
x2-10x+25-25
Add and subtract 25
Now, transform the right side into a square of a binomial, then leave y on the left side.
y-15
=
(x-5)(x-5)-25
y-15
=
(x-5)2-25
y-15+15
=
(x-5)2-25+15
y
=
(x-5)2-10
The function is now in vertex form
Compare the function to the general vertex form to get the vertex
y
=
a(x-h)2+k
y
=
(x-5)2-10
h
=
5
k
=
-10
This means that the vertex is at (5,-10)
Next, find the x intercepts by substituting y=0, then solving for x
Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.
Plot the vertex and x intercepts, then connect the points to form a parabola
First, find the vertex and plot this on the graph
Start by transforming the function into vertex form
y
=
-2x2-8x+2
y
=
-2(x2+4x-1)
Factor out -2
Take the coefficient of the x term, divide it by two and then square it.
y
=
-2(x2+4x-1)
Coefficient of the x term
=
42
Divide it by 2
(2)2
=
4
Square
This number will make the right side a perfect square.
Add and subtract 4 to the grouping of x to keep the balance.
y
=
-2(x2+4x-1)
y
=
-2(x2+4x+4-4-1)
Add and subtract 4
y
=
-2((x+2)2-4-1)
y
=
-2((x+2)2-5)
y
=
-2(x+2)2+10
This is now in vertex form
Compare the function to the general vertex form to get the vertex
y
=
a(x-h)2+k
y
=
-2(x+2)2+10
h
=
-2
k
=
10
This means that the vertex is at (-2,10)
Next, find the x intercepts by substituting y=0, then solving for x
y
=
-2(x+2)2+10
0
=
-2(x+2)2+10
Substitute y=0
0-10
=
-2(x+2)2+10-10
Subtract 10 from both sides
-10
=
-2(x+2)2
-10รท(-2)
=
-2(x+2)2รท(-2)
Divide both sides by -2
5
=
(x+2)2
(x+2)2
=
5
โ(x+2)2
=
โ5
Take the square root of both sides
x+2
=
ยฑโ5
x+2-2
=
ยฑโ5-2
Subtract 2 from both sides
x
=
-2ยฑโ5
Mark these 2 points on the x axis
Finally, connect the points to form a parabola
Question 3 of 4
3. Question
By completing the square, which graph is correct for the equation: y=x2-8x+24.
Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.
Perform the process of completing the square on the given quadratic to convert into vertex form.
y
=
x2-8x+24
=
(x2-8x+(-82)2)-(-82)2+24
Complete the square
=
(x2-8x+16)-16+24
Simplify
=
(x2-8x+16)+8
y
=
(x-4)2+8
Rewrite as a square of a binomial
Identify the vertex of the graph from the given formula.
y
=
a(xโh)2+k
y
=
(x-4)2+8
Given equation
y
=
a(xโ4)2+8
Extract values of h and k
Vertex
=
(h,k)
Vertex
=
(4,8)
Mark the vertex on the graph.
Next, solve for the y-intercept by substituting x=0.
y
=
(x-4)2+8
y
=
(0-4)2+8
Substitute x=0
y
=
16+8
y
=
24
Mark the y-intercept on the graph.
Draw a parabola using the points.
Question 4 of 4
4. Question
By completing the square, which graph is correct for the equation: y=x2-8x+18.
Completing the square is done by producing a square of a binomial on the left side of the equal sign. This method is useful when no two rational numbers solve the equation.
Perform the process of completing the square on the given quadratic to convert into vertex form.
y
=
x2-8x+18
=
(x2-8x+(-82)2)-(-82)2+18
Complete the square
=
(x2-8x+16)-16+18
Simplify
=
(x2-8x+16)+2
y
=
(x-4)2+2
Rewrite as a square of a binomial
Identify the vertex of the graph from the given formula.
y
=
a(xโh)2+k
y
=
(x-4)2+2
Given equation
y
=
a(xโ4)2+2
Extract values of h and k
Vertex
=
(h,k)
Vertex
=
(4,2)
Mark the vertex on the graph.
Next, solve for the y-intercept by substituting x=0.