Video Transcript
The equation 15𝑥 squared plus 29𝑥 minus 36 over 𝑘𝑥 minus four equals negative five 𝑥 minus three minus 48 over 𝑘𝑥 minus four is true for all values such that 𝑥 is not equal to four over 𝑘, where 𝑘 is a constant. What is the value of 𝑘?
Looking at this equation, we notice that two of the three terms have the denominator 𝑘𝑥 minus four. And so, the first thing we want to try to do is make a denominator of 𝑘𝑥 minus four in this third term. If we multiply negative five 𝑥 minus three by 𝑘𝑥 minus four over 𝑘𝑥 minus four, we haven’t changed the value. We’ve just put it in a format that’s easier to work with. We bring everything else down. And then we can multiply this entire equation by 𝑘𝑥 minus four. And what that does is get rid of the denominator in all three terms.
Our new equation says 15𝑥 squared plus 29 𝑥 minus 36 equals negative five 𝑥 minus three times 𝑘𝑥 minus four minus 48. This is because this entire expression negative five 𝑥 minus three has to be multiplied by 𝑘𝑥 minus four. Before we do anything else, we’ll have to expand this expression. Sometimes we call this foiling. We multiply the firsts, negative five 𝑥 times 𝑘𝑥, which gives us negative five 𝑘𝑥 squared. Then, we multiply the outer: negative five 𝑥 times negative four equals positive 20𝑥. The inner two terms negative three times 𝑘𝑥 equals negative three 𝑘𝑥. And the lasts, negative three times negative four equals positive 12.
We can try to add these terms and combine any like terms. Since we know that 𝑘 is a constant, 20𝑥 and negative three 𝑘𝑥 are like terms. We can combine them by calling this 20 minus three 𝑘𝑥 and then we have plus 12. So we take this expanded form and we put it back in our equation. And we now have negative five 𝑘𝑥 squared plus 20 minus three 𝑘𝑥 plus 12 minus 48. We can combine positive 12 and negative 48 to be negative 36. At this point, we’ll need to bring down what’s on the left side of the equal sign, 15𝑥 squared plus 29𝑥 minus 36. Since we have negative 36 on both sides of the equation, they cancel out.
Now we need to look at the coefficients of 𝑥 squared and the coefficients of 𝑥. Since we know that these two equations are equal to each other, these two coefficients have to be equal to each other. 15 has to be equal to negative five 𝑘 and 29 has to be equal to 20 minus three 𝑘. We can set up two new equations to say this and then we’re ready to solve for 𝑘. If 15 equals negative five 𝑘, we can divide both sides of the equation by negative five. 15 divided by negative five equals negative three. 𝑘 equals negative three.
And we can use our second equation to check and see if that’s true. We can plug in negative three for 𝑘 and then ask is 29 equal to 20 minus three times negative three. Three times negative three is positive nine. 29 is equal to 20 plus nine. That’s true. And it confirms that 𝑘 equals negative three.
