7. Solve for   x   and check

3x2 - x - 5 = 0

This will not factor, so the simplest method for solving is to use the quadratic formula

where

• a = 3,
• b = -1
• c = -5

Substitute these numbers into the formula.

Do the arithmetic under the radical: first the power, then the product.

When we finish the arithmetic under the radical, we may as well complete the calculations outside the radical.

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Check:

The checks to these equations are extremely fascinating. Let us first check the solution

Copy down the original equation

3x2 - x - 5 = 0

except that wherever you see an   x,   copy down the solution in parentheses.

This gives us an extensive order of operations problem. Since the expressions inside the parentheses are as simplified as possible, we do the powers first. There is a square in the first term. Note that the factor being squared is a fraction with a binomial on the top. When we square the top and square the bottom, we use the technique for squaring binomials.

(a + b)2 = a2 + 2ab + b2

This gives us

When we cancel the 3 and the 36, the first fraction has a denominator of 12, so we would need to multiply the top and bottom of the second fraction by 2 and the last term by 12 to get common denominators.

We could easily turn the 5 into a 60/12 to get it over a common denominator as well, but one finds that when one does these checks that if we simplify the fractions we already have at this point, it will make our lives simpler. Note how the radical terms vanish leaving us with

The fraction simplifies to

or

5 - 5 = 0

which checks.

The check of the other solution is entirely similar and is left to the reader.

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