Grades 9-10 Video Solutions 2021-video 2021 9-10/22

Video Transcription

Problem number 22 states, the numbers a, b, and c satisfy a plus b plus c equals zero, and a times b times c equals 78. What is the value of a plus b times b plus c times c plus a?  So in this problem, we are given two equations and three unknowns. So this means that it's not going to be possible to solve for a, b, and c, but that does not  mean that we cannot do some algebraic manipulation to solve for a plus b times b plus c times c plus a.  So in order to get these values, we will use first that a plus b plus c equals zero and subtract each of the variables.  This yields a plus b equals minus c, a plus c equals minus b, and b plus c equals minus a. Let's substitute these new values into our original equation.  Finally let's factor out all the negatives. Two of the three negatives cancel each other out, and we are just left with one negative.  So in the end, the product is equal to negative a, b, c, and we know the value of a, b, c. We know that a, b, c is 78. So our final answer would be minus 78.  So the question asked us, what is the value of a plus b times b plus c times c plus a? And I don't see our answer here, so the answer is none of the above, letter e.

Video Summary

The problem involves numbers \(a\), \(b\), and \(c\) satisfying \(a + b + c = 0\) and \(abc = 78\). To find the value of \((a + b)(b + c)(c + a)\), we use algebraic manipulation, substituting \(a+b = -c\), \(b+c = -a\), and \(c+a = -b\). Simplifying, we find the expression equals \(-abc\), which is \(-78\). Therefore, the solution to the expression is \(-78\). Since the answer isn't listed in the options, it is "none of the above."

Keywords

algebra

expression

manipulation

solution

none