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### Question Description

the difference between the solutions to the equation x squared = a is 30. what is a?

Wallace H
School: UCLA   I am assuming that 'x' and 'a' are the variables we are using for the solutions.

The equation representing the difference between both solutions would probably be:

a - x = 30

Since we are given that  x^2 = a, that means that   a = x^2.  And we can substitute into the equation and then solve for x.

a - x = 30                 <Difference Equation>

x^2 - x = 30               <Since  a = x^2, we can replace a with x^2>

Equation is now quadratic, so we can solve for x

x^2 - x - 30 = 30 - 30                                                    <Subtract both sides by 30>

x^2 - x - 30 = 0

(x + 5)(x - 6) = 0                                                               <Left side is factored>

x + 5 = 0             or             x - 6 = 0                                            <Zero Product Property>

x + 5 - 5 = 0 - 5      or      x - 6 + 6 = 0 + 6                           <Solve each equation separately>

x = -5                or          x = 6

I am thinking we need to use the positive x-value, so we can try x = 6

Now that we have our value of x, we can substitute back into our difference equation to solve for 'a'

a - x = 30               <Difference Equation>

a - 6 = 30               <Substituting our calculated value of x, which is x = 6>

a - 6 + 6 = 30 + 6      <Add both sides by 6>

a = 36

POSSIBLE SOLUTION:    a = 36                (Only if you are required to use just the positive x value)

I don't know if your problem requires more than one answer.  But just in case, we will also test  x = -5

a - x = 30                  <Difference Equation>

a - (-5) = 30                <Substitute x = -5>

a + 5 = 30                   < A - (-B) is the same as A + B>

a + 5 - 5 = 30 - 5         <Subtract both sides by 5>

a = 25

POSSIBLE SOLUTION:       a = 36  or   a = 25                   (Only if you needed more than one solution)

I hope this helps.

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