If four 𝑦 minus three multiplied
by five 𝑦 plus six equals 20𝑦 squared plus 𝑘𝑦 minus 18, what is the value of
On the left-hand side of this
equation, we have the product of two binomials, four 𝑦 minus three and five 𝑦 plus
six. On the right-hand side, we’re told
what this product is equal to in its expanded form. But this expression involves the
unknown 𝑘, the value of which we need to determine.
To do so, let’s begin by expanding
the product ourselves. We must ensure that we multiply
each term in the first binomial by each term in the second. So we should initially obtain four
terms. Multiplying the first terms in each
binomial together gives 20𝑦 squared. Multiplying the terms on the
outside of the product together gives 24𝑦. Multiplying the terms on the inside
of the product together gives negative 15𝑦. And finally, multiplying the last
terms in each binomial together gives negative 18. We have four terms as expected. But we can now simplify this
expression by collecting the like terms. That gives 20𝑦 squared plus nine
𝑦 minus 18.
We can now compare this expansion
with the expression we were given on the right-hand side of the equation. In doing so, we see that the
coefficient of 𝑦 squared and the constant term are the same on each side of the
equation, as they should be. In order for the equation to be
true for every value of 𝑦, it must also be the case that the final coefficient, the
coefficient of 𝑦, is the same on both sides of the equation. Equating these coefficients gives
𝑘 equals nine.
So, by expanding the product of
these two binomials ourselves and equating the coefficients of 𝑦, we’ve found that
the value of 𝑘 is nine.