What is Completing the Square?
ie: Re-write a quadratic in the form:
Worked Examples:
x² + 8x ≡ (x+p)² + q
x² + 8x ≡ x² + 2px + p² = q Compare coefficients: Coefficient: LHS : RHS x² 1 : 1 x 8 : 2p Therefore: p = 4 Const. 0 0 : p² + q 0 : 16 + q Therefore: q = -16 Therefore: x² + 8x = (x+4)² - 16 |
x² + 6x + 10 ≡ (x+p)² + q
p = 6/2 = 3 q = 10 - p² q = 10 - 9 q = 1 Therefore: x² + 6x + 10 = (x+3)² + 1 |
Deriving the Formula:
x² + bx + c ≡ (x+p)² + q
p = b/2
q = c - (b/2)²
Therefore:
x² + bx + c ≡ (x+b/2)² + (c-(b/2)²)
p = b/2
q = c - (b/2)²
Therefore:
x² + bx + c ≡ (x+b/2)² + (c-(b/2)²)
Why Complete the Square?
1. You can use completing the square to solve equations
EG: m² - 3m + 1 = 0 - cannot be factorised
So therefore, we have to complete the square:
(m-3/2)² - 5/4 = 0
Now solve:
(m-3/4)² = 5/4
m-3/4 = +/- √5/4
m = 3/4 +/- √5/4
2. You can also use completing the square to calculate the coordinates of the Vertex of the graph:
EG: y = (x+p)² + q
Coordinates of the Vertex = (-p, q)
EG: m² - 3m + 1 = 0 - cannot be factorised
So therefore, we have to complete the square:
(m-3/2)² - 5/4 = 0
Now solve:
(m-3/4)² = 5/4
m-3/4 = +/- √5/4
m = 3/4 +/- √5/4
2. You can also use completing the square to calculate the coordinates of the Vertex of the graph:
EG: y = (x+p)² + q
Coordinates of the Vertex = (-p, q)
Worked Examples:
Complete the square and hence solve the equation:
y² - 4y + 2 = 0 (y-2)² -2 = 0 (+2) (y-2)² = 2 (square root) y-2 = +/- √2 y = 2 +/- √2 Complete the square and hence solve the equation: x² - 2y = 11 (x-1)² = 11 (-11) (x-1)² - 11 = 0 q = -1 (x-1)² - 11 - 1 = 0 (x-1)² - 12 = 0 (+12) (x-1)² = 12 (√) x-1 = +/- √12 (+1) y = 1 +/- √12 |
Complete the square and hence solve the equation:
u² + 7u = 44 (u+7/2)² = 44 (-44) (u+7/2)² - 44 = 0 q = - 44 - (7/2)² q = - 44 - 49/4 q = -176/4 - 49/4 q = -215/4 (u+7/2)² - 215/4 = 0 (+215/4) (u+7/2)² = 215/4 (√) u+7/2 = +/- √215/4 (-7/2) u = -7/2 =/- √215/4 |
Completing the Square with Coefficients:
ie: Re-write a quadratic in the form:
Worked Examples:
2x² - 8x + 7 ≡ r(x+p)² + q
2x² - 8x + 7 ≡ r(x+p)(x+p) + q 2x² - 8x + 7 ≡ r(x²+2p+p²) + q 2x² - 8x + 7 ≡ rx² + 2prx + (p²)r + q Compare coefficients: Coefficient: LHS : RHS x² 2 : r x -8 : 2pr Therefore: p = -2 Const. 7 : p² + q 7 : -2² x 2 + q Therefore: q = -16 7 : 8 + q Therefore: q = -1 Therefore: 2x² - 8x + 7 ≡ 2(x-2)² - 1 |
3x² - 9x + 5 ≡ r(x+p)² + q
r = 3 p = -9/(2x3) = -9/6 = -3/2 q = 5 - (-3/2)² x 3 q = 5 - 9/4 x 3 q = 5 - 27/4 q = 20/4 - 27/4 q = -7/4 Therefore: 3x² - 9x + 5 = 3(x-3/2)² - 7/4 |
Deriving the Formula:
ax² + bx + c ≡ r(x+p)² + q
r = a
p = b/2a
q = c - ((b/2a)²)a
Therefore:
ax² + bx + c ≡ a(x+b/2a)² + (c - ((b/2a)²)a)
r = a
p = b/2a
q = c - ((b/2a)²)a
Therefore:
ax² + bx + c ≡ a(x+b/2a)² + (c - ((b/2a)²)a)
Worked Examples:
Complete the square for the equation: 2x² + 4x + 3
r = 2 p = 4/(2x2) = 4/4 = 1 q = 3 - (1)² x 2 q = 3 - 2 q = 1 Therefore: 2x² + 4x + 3 = 2(x+1)² + 1 |
Complete the square for the equation: -x² - 2x - 5
r = -1 p = -2/(2x-1) = -2/-2 = 1 q = -5 - (1)² x -1 q = -5 - -1 q = -4 Therefore: -x² - 2x - 5 = -1(x+1)² - 4 |