Question Video: Multiplying Two Binomials to Find the Value of an Unknown Mathematics • 9th Grade

If (4𝑦 − 3)(5𝑦 + 6) = 20𝑦² + 𝑘𝑦 − 18, what is the value of 𝑘?

02:18

Video Transcript

If four 𝑦 minus three multiplied by five 𝑦 plus six equals 20𝑦 squared plus 𝑘𝑦 minus 18, what is the value of 𝑘?

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.

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