Find the exact roots of 0= x^2 -7x +7
I think you have to use the quadratic formula, but I can't figure out how to do this or how to verify. If you could show me step by step, that would be awesome. Thanks
Every student of mathematics must know and memorize the quadratic equation formula:
for ax^2 + bx + c = 0
x = (-b ± √(b^2 - 4ac) /(2a)
for yours:
a = 1
b = -7
c = 7
x = (7 ± √(49 - 4(1)(7))/(2
= (7 ± √21)/2
x = (7 + √21)/2 OR x = (7 - √21)/2
Since it asked for the "exact" roots, you leave the answers that way.
If you are asked to "verify" , sub those values into the original equation. I will do one of the them , you try the other
if x = (7+√21)/2
LS = [(7+√21)/2]^2 - 7(7+√21)/2 + 7
= (49 + 14√21 + 21)/4 - (49 + 7√21)/2 + 7
= (70 + 14√2 - 98 - 14√21 + 28)/4
= 0/4
= 0
= RS
so x = (7 + √21)/2
You try the 2nd root.
Notice the verification looked messier than the original solving.
=
To find the exact roots of the quadratic equation 0 = x^2 - 7x + 7, you are correct that the quadratic formula can be used. The quadratic formula states that for any quadratic equation in the form ax^2 + bx + c = 0, the solutions (or roots) can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the coefficients are a = 1, b = -7, and c = 7. Now, let's go step by step to find the roots:
Step 1: Identify the coefficients a, b, and c.
a = 1
b = -7
c = 7
Step 2: Apply the quadratic formula to find the roots. Substitute the values for a, b, and c into the formula:
x = (-(-7) ± √((-7)^2 - 4(1)(7))) / (2(1))
Simplifying:
x = (7 ± √(49 - 28)) / 2
x = (7 ± √21) / 2
So, we have two possible solutions:
x₁ = (7 + √21) / 2
x₂ = (7 - √21) / 2
These are the exact roots of the equation 0 = x^2 - 7x + 7. To verify if they are the correct roots, substitute them back into the original equation and see if both sides equal zero.