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
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Teacher
To solve the quadratic equation 9x^2 + 9x + 4 = 0 by completing the square, follow these steps:
Step 1: Move the constant term (4) to the right side of the equation:
9x^2 + 9x = -4
Step 2: Divide the entire equation by the coefficient of x^2 (9) to make the coefficient of x^2 equal to 1:
x^2 + x = -4/9
Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 + x + (1/2)^2 = -4/9 + (1/2)^2
x^2 + x + 1/4 = -4/9 + 1/4
Step 4: Simplify the equation on the right side:
x^2 + x + 1/4 = -4/9 + 2/8
x^2 + x + 1/4 = -32/72 + 18/72
x^2 + x + 1/4 = -14/72
x^2 + x + 1/4 = -7/36
Step 5: Rewrite the left side of the equation as a perfect square:
(x + 1/2)^2 = -7/36
Step 6: Take the square root of both sides:
√((x + 1/2)^2) = ± √(-7/36)
x + 1/2 = ± √(-7/36)
Step 7: Simplify the square root on the right side:
x + 1/2 = ± (√(-7)/√(36))
x + 1/2 = ± (√(-7)/6)
x + 1/2 = ± (√7 i/6) (where i is the imaginary unit)
Step 8: Subtract 1/2 from both sides of the equation:
x = -1/2 ± (√7 i/6)
The solutions to the quadratic equation 9x^2 + 9x + 4 = 0, when solved by completing the square, are:
x = -1/2 - (√7 i/6), -1/2 + (√7 i/6)
Therefore, the answers for blank 1, blank 2, and blank 3 are:
Blank 1: 1/4
Blank 2: 7
Blank 3: √7/6