Video Transcript
In the equation, two 𝑚 squared
plus three 𝑚 minus nine over 𝑚 squared plus seven 𝑚 plus 12 equals 𝑚 squared
minus six 𝑚 plus five over 𝑚 squared minus 𝑚 minus 20. What is the value of 𝑚?
In this question, we have four
different polynomials. The best way to solve this is to
factor each of these polynomials and see if there’s anything we can cancel out
before we try to solve for 𝑚. First we’ll factor the polynomial
on the top left. Since we have a first term of two
𝑚 squared, we’ll break that up into two 𝑚 and 𝑚. And then we need two values that
multiply together to equal negative nine and that add together to equal positive
three.
I know that three times three
equals nine. Since we have negative nine, we
need a positive three and a negative three. And two 𝑚 times three equals
positive six 𝑚. Negative three times 𝑚 equals
negative three 𝑚. And so we get our positive three 𝑚
we need. And now, we factor the
denominator. We break up the 𝑚 squared as 𝑚
times 𝑚. We need values that multiply
together to equal 12 and add together to equal seven. Four times three multiplies
together to equal 12 and adds together to equal positive seven.
From there, we factor the numerator
on the second side, 𝑚 squared. We break that apart to be 𝑚 and
𝑚. We’re looking for values that
multiply together to equal five and added together equals negative six. Five only has two factors, five and
one. We need these values to add
together to equal negative six. That means both the five and the
one should be negative. Negative five times negative one
still equals positive five. We need to factor our final
polynomial. 𝑚 squared will factor as 𝑚 and
𝑚. Our terms need to multiply together
to equal negative 20 and add together to equal negative 𝑚. Or in this case, we’re looking for
negative one.
I know that five times four equals
20 and that negative five plus four would equal negative one. Positive four times negative five
equals negative 20. And so we have our final two
terms. And at this point, we start to see
some things that cancel out. We have an 𝑚 plus three in the
numerator and the denominator on the left. And we have 𝑚 minus five in the
numerator and the denominator on the right.
We simplify this to say two 𝑚
minus three over 𝑚 plus four equals 𝑚 minus one over 𝑚 plus four. If we multiplied both sides of the
equation by 𝑚 plus four, the denominators would cancel out, so that we have two 𝑚
minus three equals 𝑚 minus one. To get the 𝑚s on the same side, we
subtract 𝑚 from both sides. 𝑚 minus three equals negative
one. And so we add three to both
sides. And we find that 𝑚 equals two.
