This is like the preceding problem, except that your answers may be negative
x^2+7/3x+ =(x+ )^2.
x^2−5/7x+ =(x+ )^2.
x^2+2/3x+ =(x+ )^2.
To find the missing terms (represented by empty parentheses) and complete the square for each quadratic equation, we can follow these steps:
Step 1: Write the quadratic equation in the standard form: ax^2 + bx + c = 0
For the first equation:
x^2 + (7/3)x + ___ = (x + ___)^2
For the second equation:
x^2 - (5/7)x + ___ = (x + ___)^2
For the third equation:
x^2 + (2/3)x + ___ = (x + ___)^2
Step 2: Identify the coefficient of the x-term (b) and divide it by 2. This will give us the value to complete the square.
For the first equation:
b = 7/3, so (7/3) / 2 = 7/6
For the second equation:
b = -5/7, so (-5/7) / 2 = -5/14
For the third equation:
b = 2/3, so (2/3) / 2 = 1/3
Step 3: Square the value obtained in step 2 to get the term that will complete the square.
For the first equation:
(7/6)^2 = 49/36
For the second equation:
(-5/14)^2 = 25/196
For the third equation:
(1/3)^2 = 1/9
Step 4: Add the term obtained in step 3 to both sides of the equation.
For the first equation:
x^2 + (7/3)x + 49/36 = (x + ___)^2 + 49/36
For the second equation:
x^2 - (5/7)x + 25/196 = (x + ___)^2 + 25/196
For the third equation:
x^2 + (2/3)x + 1/9 = (x + ___)^2 + 1/9
Step 5: Complete the square by factoring the square of the binomial on the left side.
For the first equation:
(x + 7/6)^2 = (x + ___)^2 + 49/36
For the second equation:
(x - 5/14)^2 = (x + ___)^2 + 25/196
For the third equation:
(x + 1/3)^2 = (x + ___)^2 + 1/9
By comparing the equations, we can see that for each equation, the missing term inside the parentheses is equal to half of the coefficient of the x-term.
For the first equation: x^2 + (7/3)x + 49/36 = (x + 7/6)^2
For the second equation: x^2 - (5/7)x + 25/196 = (x - 5/14)^2
For the third equation: x^2 + (2/3)x + 1/9 = (x + 1/3)^2
Therefore, the missing terms are:
For the first equation: 7/6
For the second equation: -5/14
For the third equation: 1/3