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Mathematics

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