Solve a quadratic equation by completing the square
Key Notes :
Understanding Completing the Square
- Completing the square is a technique used to rewrite a quadratic equation in the form (x + h)² = k, where h and k are constants.
- This form makes it easier to solve for x by taking the square root of both sides.
Steps to Complete the Square
- Ensure the Leading Coefficient is 1: If the leading coefficient (a) is not 1, divide the entire equation by a to make it 1.
- Isolate the Quadratic and Linear Terms: Move the constant term (c) to the other side of the equation.
- Add the Square of Half the Linear Coefficient: Add (b/2)² to both sides of the equation. This will create a perfect square trinomial on the left side.
- Factor the Perfect Square Trinomial: The left side of the equation should now be in the form (x + h)².
- Solve for x: Take the square root of both sides and solve for x.
Example
Solve: x² – 6x + 2 = 0
- Step 1: The leading coefficient is already 1.
- Step 2: Move the constant term: x² – 6x = -2
- Step 3: Add the square of half the linear coefficient: x² – 6x + 9 = -2 + 9
- Step 4: Factor the perfect square trinomial: (x – 3)² = 7
- Step 5: Take the square root: x – 3 = ±√7
- x = 3 ± √7
Key Points
- Completing the square is a useful technique for solving quadratic equations, especially when factoring is not straightforward.
- The goal is to create a perfect square trinomial on the left side of the equation.
- Adding the square of half the linear coefficient is the key step in completing the square.
- Once the equation is in the form (x + h)² = k, solving for x is straightforward by taking the square root.
Learn with an example
Solve by completing the square.
s2 + 22s = 1
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.
s = ________ or s = ________
Step 1: Make sure that the left side of the equation looks like x2 + bx.
This step does not apply, because the left side of the equation already looks like x2 + bx.
s2 + 22s = 1
Step 2: Add (b/2)2 to both sides.
Since b=22,(b/2)2 =(22/2)2 =112 =121.Add 121 to both sides.
s2 + 22s + 121 = 122.
Step 3: Factorise the left side as (x + b/2)2 .
In general, an expression of the form x2 + bx +( b/2)2 can be factorised as (x + b/2)2 .
The expression s2 + 22s + 121 is of this form,with b=22. So, it can be factorised as (s + 11)2.
Rewrite the equation with the left side factorised.
(s + 11)2 = 122
Step 4: Take the square root and solve.
s + 11 ≈ ±11.05 Take the square root
s ≈ -11 ± 11.05 Subtract11frombothsides
s ≈ -11 + 11.05 or s ≈ -11 − 11.05 Split ± into + or –
s ≈ 0.05 or s ≈ -22.05 Simplify
Solve by completing the square.
s2 + 6s − 11 = 0
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.
s = ________or s =________
Step 1: Make sure that the left side of the equation looks like x2 + bx.
To make the left side of the equation look like x2 + bx, add 11 to both sides.
s2 + 6s − 11 = 0
s2 + 6s = 11
Step 2: Add (b/2)2 to both sides.
Since b=6, (b/2)2 = (6/2)2 = 32 = 9.Add 9 to both sides.
s2 + 6s + 9 = 20
Step 3: Factorise the left side as (x + b/2)2 .
In general, an expression of the form x2 + bx + (b/2)2 can be factorised as (x + b/2)2 .
The expression s2 + 6s + 9 is of this form,with b=6.So,it can be factorised as (s + 3)2.
Rewrite the equation with the left side factorised.
(s + 3)2 = 20
Step 4: Take the square root and solve.
s + 3 ≈ ±4.47 Take the square root
s ≈ -3 ± 4.47 Subtract3 from both sides
s ≈ -3 + 4.47 or s ≈ -3 − 4.47 Split ± into + or –
s ≈ 1.47 or s ≈ -7.47 Simplify
Solve by completing the square.
q2 + 22q = 13
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.
q = ________or q = ________
Step 1: Make sure that the left side of the equation looks like x2 + bx.
This step does not apply, because the left side of the equation already looks like x2 + bx.
q2 + 22q = 13
Step 2: Add (b/2)2 to both sides.
Since b=22, (b/2)2 = (22/2)2 = 112 = 121.Add 121 to both sides.
q2 + 22q + 121 = 134
Step 3: Factorise the left side as (x + b/2)2 .
In general, an expression of the form x2 + bx +( b/2)2 can be factorised as x + (b/2)2 .
The expression q2+ 22q + 121 is of this form, with b=22. So,it can be factorised as (q + 11)2.
Rewrite the equation with the left side factorised.
(q + 11)2 = 134
Step 4: Take the square root and solve.
q + 11 ≈ ±11.58 Take the square root
q ≈ -11 ± 11.58 Subtract 11from both sides
q ≈ -11 + 11.58 or q ≈ -11 − 11.58 Split ± into + or –
q ≈ 0.58 or q ≈ -22.58 Simplify
let’s practice!
