document.write( "Question 549064: Solve X^2 + 11 = 6x \n" ); document.write( "
Algebra.Com's Answer #357454 by KMST(5328)\"\" \"About 
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\"x%5E2%2B11=6x\" is a quadratic equation.
\n" ); document.write( "In the (equivalent) traditional form to present a quadratic equation, it would be
\n" ); document.write( "\"x%5E2-6x%2B11=0\"
\n" ); document.write( "SHORT ANSWER:
\n" ); document.write( "The discriminant \"b%5E2-4ac=%28-6%29%5E2-4%2A1%2A11=36-44=-4\" is negative, so there are no real solutions.
\n" ); document.write( "LONG GENERAL EXPLANATION ON QUADRATIC EQUATIONS:
\n" ); document.write( "There are many approaches to solve quadratic equations:
\n" ); document.write( "You could factor, graph, apply the quadratic formula, or \"complete the square,\"
\n" ); document.write( "Factoring does not work all the time. It only works when the solutions are rational numbers (not the case for this equation) and, even in cases where it would work, it is an approach that takes some practice.
\n" ); document.write( "Graphing adds to your understanding, and it will work sometimes, but it is useful mainly if you have a graphing calculator or other graphing software, and the exact answer is not required.
\n" ); document.write( "QUADRATIC FORMULA:
\n" ); document.write( "Using the quadratic formula works every time, without needing to think much, but requires using the complicated looking quadratic formula:
\n" ); document.write( "\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"
\n" ); document.write( "The expression under the square root is called the discriminant, and it is good to calculate that part first.
\n" ); document.write( "\"discriminant=b%5E2-4%2Aa%2Ac\"
\n" ); document.write( "After that, the quadratic formula simplifies into
\n" ); document.write( "\"x=%28-b+%2B-+sqrt%28discriminant%29%29%2F%282%2Aa%29\"
\n" ); document.write( "You can see that the quadratic formula will not give you a real solution if the discriminant is negative. You would answer that there are no real solutions.
\n" ); document.write( "You can see that the quadratic formula simplifies into \"x+=-b+%2F%282%2Aa%29\" when the discriminant is zero.
\n" ); document.write( "If the discriminant is positive, you will get two real solutions.
\n" ); document.write( "You can identify the values for a, b, and c, by comparing the coefficients in your equation to those in the generic quadratic equation
\n" ); document.write( "\"ax%5E2%2Bbx%2Bc=0\"
\n" ); document.write( "The coefficient of the term in \"x%5E2\" is a. If it's just \"x%5E2\" with no visible coefficient, \"a=1\". If it's just \"-x%5E2\", then \"a=-1\".
\n" ); document.write( "The coefficient of x is b. If nothing is visible in front of an x, b could be 1, or -1, as with a. If you don't even see a term in x, \"b=0\".
\n" ); document.write( "The independent term (the term without x) is c. If you don't see an independent term, then \"c=0\".
\n" ); document.write( "For \"x%5E2-6x%2B11=0\", a=1, b=-6, and c=11
\n" ); document.write( "The discriminant, \"b%5E2-4ac=%28-6%29%5E2-4%2A1%2A11=36-44=-4\", is negative, so there are no real solutions. (If you are allowed to work with imaginary numbers, you could calculate two complex solutions using the quadratic formula).
\n" ); document.write( "COMPLETING THE SQUARE
\n" ); document.write( "\"Completing the square\" takes some practice, and could be difficult at times, but it does not require memorizing formulas, and would work easily in this case. I'll demonstrate.
\n" ); document.write( "We can write the equation with the terms containing x on one side of the equal sign, and the term without x on the other side:
\n" ); document.write( "\"x%5E2-6x%2B11=0\" ---> \"x%5E2-6x=-11\"
\n" ); document.write( "At this point, we try to imagine \"x%5E2-6x\" as part of the square of a binomial and see that
\n" ); document.write( "\"x%5E2-6x%2B9=%28x-3%29%5E2\"
\n" ); document.write( "We add 9 to both sides of the equation \"x%5E2-6x=-11\" to get
\n" ); document.write( "\"x%5E2-6x%2B9=-11%2B9\" <---> \"%28x-3%29%5E2=-2\"
\n" ); document.write( "There is no happy ending here, because at this point we realize that there are no real solutions, because the square of a real number is never negative.
\n" ); document.write( "If we are allowed to work with imaginary numbers, we can continue on to \"x=3+%2B-+isqrt%282%29\". Otherwise, we just say that there are no real solutions.\r
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