Answer:
1) x = 6 + 2√19 or x = 6 - 2√19
2a) (x + 5)²
2b) (x + 5/2)²
3) x = 22 or 18
Step-by-step explanation:
1) x² - 12x = 40
First step is;
We take ½ of the x-term and square it.
Thus;
(½ × -12)² = (-6)² = 36
Second step is to add the result in step 1 to both sides;
x² - 12x + 36 = 40 + 36
x² - 12x + 36 = 76
On the left, is a perfect square that can be written as (x - 6)²
Thus, the new equation is;
(x - 6)² = 76
Taking square root of both sides gives;
x - 6 = ±√76
x - 6 = ±2√19
x = 6 + 2√19 or x = 6 - 2√19
2a) we want to find the value of c that makes x² + 10x + c the perfect square.
From completing the square in question 1 above, we see that;
c = (b/2)²
From ax² + bx + c, we can see that in this case, b = 10
Thus;
c = (10/2)²
c = 5²
c = 25
Thus, the trinomial is;
x² + 10x + 25
As a perfect square, it is;
(x + 5)²
2b) x² + 5x + c
Similar to 2a above;
b = 5
Thus; c = (5/2)²
c = 25/4
Trinomial is; x² + 5x + 25/4
As a perfect square, we have;
(x + 5/2)²
3) We want to solve x² - 40x + 396 = 0
This can be rewritten as;
x² - 40x = -396
First step is;
We take ½ of the x-term and square it.
Thus;
(½ × -40)² = 400
Second step is to add the result in step 1 to both sides;
x² - 40x + 400 = -396 + 400
x² - 40x + 400 = 4
On the left, is a perfect square that can be written as (x - 20)²
Thus;
(x - 20)² = 4
Taking square root of both sides gives;
x - 20 = ±√4
x - 20 = ±2
x = 20 + 2 or x = 20 - 2
x = 22 or 18
