Part A
Get everything to one side. The quickest way to do this is to add 1 to both sides
x^2 - 6x = -1
x^2 - 6x + 1 = -1+1
x^2 - 6x + 1 = 0
The equation is now in the form ax^2 + bx + c = 0
We see that,
a = 1
b = -6
c = 1
Plug those values into the discriminant formula
d = b^2 - 4ac
d = (-6)^2 - 4*1*1
d = 36 - 4
d = 32
The value of the discriminant is 32
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Part B
In part A, we got d = 32 as the discriminant. The discriminant is positive (greater than 0) so there are 2 different real number solutions. These solutions are irrational because d = 32 is not a perfect square. Taking the square root of 32 leads to a decimal value that goes on forever without any known pattern. It is impossible to represent sqrt(32) as a fraction of two whole numbers.
In short, the two solutions are irrational.
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Part C
Use the quadratic formula to solve for x. Check out the image attachment for more detailed steps. The two irrational solutions, rounded to the nearest tenth, are x = 0.2 or x = 5.8
Get everything to one side. The quickest way to do this is to add 1 to both sides
x^2 - 6x = -1
x^2 - 6x + 1 = -1+1
x^2 - 6x + 1 = 0
The equation is now in the form ax^2 + bx + c = 0
We see that,
a = 1
b = -6
c = 1
Plug those values into the discriminant formula
d = b^2 - 4ac
d = (-6)^2 - 4*1*1
d = 36 - 4
d = 32
The value of the discriminant is 32
-------------------------------------------
Part B
In part A, we got d = 32 as the discriminant. The discriminant is positive (greater than 0) so there are 2 different real number solutions. These solutions are irrational because d = 32 is not a perfect square. Taking the square root of 32 leads to a decimal value that goes on forever without any known pattern. It is impossible to represent sqrt(32) as a fraction of two whole numbers.
In short, the two solutions are irrational.
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Part C
Use the quadratic formula to solve for x. Check out the image attachment for more detailed steps. The two irrational solutions, rounded to the nearest tenth, are x = 0.2 or x = 5.8