Answer:
Finding the vertex by completing the square
1) The vertex = (4/3, -4/3)
2) The vertex = (-1/2, 36)
3) The vertex = (2/3, 26/3)
Finding the value of x by completing the square
1) x = 1 or -4
2) x = 3/2 or -7/2
3) x = 1 or -3
Step-by-step explanation:
Finding the vertex by completing the square
The vertex form of the quadratic equation y = ax² + bx + c is written as follows;
y = a(x - h) + k
Therefore, we have;
Y = 3·x² + 4 - 8·x = a(x - h)² + k
Hence, a = 3
Which gives;
3×(x² - 2·x·h + h²) + k
Hence, we have 3×2×h = 8
h = 8/6 = 4/3
The constant term = k
We note that 3 × h² + k = 4
∴ k = 4 - 16/3 = -4/3
The vertex form is therefore;
Y = 3(x - 4/3)² - 4/3
The vertex = (4/3, -4/3)
2) Where Y = 4·x² + 4·x + 36
a = 4
4× -2·h = 4
∴ h = 4/(4 × (-2)) = 4/-8 = -1/2
Also, 4 × h² + k = 36
Which gives;
4 × (-1/2)² × k = 36
k = 36
The vertex form becomes
y = 4(x - (-1/2)) + 36
The vertex = (-1/2, 36)
3) Where Y = -3·x² + 4·x + 10, we have
a = -3
3×2×h = 4
∴ h = 4/(2×3) = 2/3
a×h² + c = k
-(3)×(2/3)² + 10 = 26/3
Which gives;
Y = -3(x - 2/3) + 26/3
The vertex = (2/3, 26/3)
To find the value of x by completing the square, we have;
1) 2·x² = -6·x + 8
2·x² + 6·x = 8
2·(x² + 3·x) = 8
x² + 3·x = 8/2 = 4
x² + 3·x + (3/2)² = 4 + (3/2)² = 25/4
(x + 3/2)² = 25/4
∴ x + 3/2 = ±5/2
x = 5/2 - 3/2 or -5/2 - 3/2
x = 1 or -4
2) 8·x² + 16·x = 42
8·(x² + 2·x) = 42
x² + 2·x = 42/8 = 21/4
x² + 2·x + 1 = 21/4 + 1 = 25/4
(x + 1)² = 25/4
x + 1 = √(25/4) = ± 5/2
x = 5/2 - 1 or -5/2 - 1
∴ x = 3/2 or -7/2
3) -x² + 2·x = -3
-x² + 2·x = -3
-1×(-x² + 2·x) = -1 ×-3 = 3
x² - 2·x = 3
x² - 2·x + 1 = 3 + 1 = 4
(x - 1)² = 4
x - 1 = √4 = ±2
∴ x = 2 - 1 or -2 - 1
x = 1 or -3
