Introduction to quadratic equations?
If you solve the equation by completing the square, fill in the blanks.
9x^2+9x+4=0
x^2+x+blank=-4/9+blank <---- (blank one, both the same)
(x+1/2)^2=blank
x+1/2= ± √blank <------ blank 2(both the same)
x=-1/2 ±blank i <---- blank 3
enter blank 1, blank 2, blank 3 as common fractions separated by commas
x^2 + x + 1/4 = -4/9 + 1/4
(x+1/2)^2 = -7/36
x+1/2 = ± √(-7/36)
x = -1/2 ± √7/6 i
Andy wants the greatest possible harvest from his apple orchard. The orchard contains 220 dwarf trees per acre now, and Andy harvests, on the average, 1300 apples per tree. Because his trees are planted in rows of 10, he adds or removes trees only in groups of 10. Andy's research has shown that for every additional 10 trees he plants per acre, the average yield per tree in the orchard will decrease by 50 apples. How many trees per acre should the orchard contain to produce the maximum harvest?
To understand the introduction to quadratic equations, consider the example equation 9x^2 + 9x + 4 = 0.
To solve this equation by completing the square, fill in the blanks step-by-step:
1. First, divide the entire equation by the leading coefficient, which is 9, to make the coefficient of x^2 equal to 1:
x^2 + x + 4/9 = 0
2. Now, take half of the coefficient of x, which is 1/2, and square it:
(1/2)^2 = 1/4
3. Add the resulting value to both sides of the equation:
x^2 + x + 4/9 + 1/4 = 1/4
4. Simplify the left side by finding a common denominator:
(36x^2 + 36x + 16)/36 + (9x^2 + 9x + 4)/36 = 1/4
Combining the numerators:
(36x^2 + 36x + 9x^2 + 9x + 16 + 4)/36 = 1/4
Simplifying the numerator:
(45x^2 + 45x + 20)/36 = 1/4
5. Multiply both sides of the equation by 36 to eliminate the fractions:
45x^2 + 45x + 20 = 9
6. Move the constant term to the right side of the equation:
45x^2 + 45x = 9 - 20
45x^2 + 45x = -11
7. Now, complete the square by taking half of the coefficient of x, which is 45/2, and squaring it:
(45/2)^2 = 2025/4
8. Add the resulting value to both sides of the equation:
45x^2 + 45x + 2025/4 = -11 + 2025/4
9. Simplify the right side:
-11 + 2025/4 = -44/4 + 2025/4
-11 + 2025/4 = 2013/4
10. Simplify the equation:
45x^2 + 45x + 2025/4 = 2013/4
11. Multiply both sides by 4 to eliminate the fractions:
180x^2 + 180x + 2025 = 2013
12. Move the constant term to the right side:
180x^2 + 180x = 2013 - 2025
180x^2 + 180x = -12
13. Divide the entire equation by the leading coefficient, which is 180, to make the coefficient of x^2 equal to 1:
x^2 + x = -12/180
x^2 + x = -1/15
14. Lastly, rewrite the equation:
x^2 + x + (1/2)^2 = -1/15 + (1/2)^2
x^2 + x + 1/4 = -1/15 + 1/4
Now fill in the blanks:
blank 1: 1/4
blank 2: -1/15 + 1/4
blank 3: ±√(blank 2) i
Therefore, the completed quadratic equation is:
(x + 1/2)^2 = 1/4
x + 1/2 = ± √( -1/15 + 1/4 )
x + 1/2 = ± √( 3/20 ) i
x = -1/2 ± √( 3/20 ) i
Thus, blank 1 = 1/4, blank 2 = -1/15 + 1/4, and blank 3 = √( 3/20 ).
To solve the quadratic equation 9x^2 + 9x + 4 = 0 by completing the square, let's go step by step:
Step 1: Divide the entire equation by the leading coefficient (9 in this case) to simplify it. The equation becomes:
x^2 + x + 4/9 = 0
Step 2: Move the constant term (4/9) to the other side of the equation, so it becomes:
x^2 + x = -4/9
Step 3: Take half of the coefficient of x (which is 1/2) and square it. This value will be added to both sides of the equation in order to complete the square. The squared value is (1/2)^2 = 1/4.
x^2 + x + 1/4 = -4/9 + 1/4
Step 4: Simplify the right side of the equation:
-4/9 + 1/4 = -16/36 + 9/36 = -7/36
Therefore, the equation becomes:
x^2 + x + 1/4 = -7/36
Step 5: Rewrite the left side of the equation as a perfect square. In this case, we have (x + 1/2)^2 = -7/36.
Therefore, blank one (for completing the square) is 1/4.
Step 6: Take the square root of both sides of the equation:
√((x + 1/2)^2) = ±√(-7/36)
Step 7: Simplify the right side of the equation (the square root):
±√(-7/36) = ±(√(-7)/√(36)) = ±(√(-7)/6)
Step 8: Therefore, the equation becomes:
x + 1/2 = ±(√(-7)/6)
Step 9: Move 1/2 to the other side of the equation:
x = -1/2 ± (√(-7)/6)
Step 10: Simplify the expression (√(-7)/6) by rationalizing the denominator:
(√(-7)/6) * (√(6)/√(6)) = (√(-42)/√(36)) = (√(-42)/6)
Therefore, blank two (in Step 8) is -7 and blank three (in Step 9) is √(-42)/6.
So, the final answer is:
Blank one: 1/4
Blank two: -7
Blank three: √(-42)/6