Video Transcript
Given that 𝑎𝑥 plus two times three 𝑥 plus 𝑏 equals three 𝑎 𝑥 squared plus nine 𝑥 plus two 𝑏 and 𝑎 plus 𝑏 equals four, find two possible values of 𝑏.
Starting with the first equation we’re given, we want to expand these two factors. We can do that by foiling, multiplying the first, outer, inner, and then last terms. 𝑎𝑥 times three 𝑥 equals three 𝑎 𝑥 squared. 𝑎𝑥 times 𝑏 equals 𝑎𝑏𝑥. Two times three 𝑥 equals six 𝑥. And two times 𝑏 equals two 𝑏. All of which is equal to three 𝑎 𝑥 squared plus nine 𝑥 plus two 𝑏. Since we have a three 𝑎 𝑥 squared on both sides of the equation, they cancel. And since we have two 𝑏 on both sides of the equation, they also cancel. And now we have 𝑎𝑏𝑥 plus six 𝑥 equals nine 𝑥. We can subtract six 𝑥 from both sides of the equation. And 𝑎𝑏𝑥 is equal to three 𝑥. If we divide both sides of the equation by 𝑥, the 𝑥s cancel out. And we’re left with the statement that says 𝑎𝑏 equals three. 𝑎 times 𝑏 must equal three. And we’ve been given that 𝑎 plus 𝑏 must equal four.
At this point, we can think about the values that multiply together to equal three and add together to equal four. One plus three equals four. And one times three equals three. If 𝑎 equals one, then 𝑏 is equal to three. And if 𝑎 equals three, then 𝑏 must be equal to one. 𝑏 equals three or 𝑏 equals one.
I also want to show you one other way to solve this problem. You might use this method if the values in these two positions were not as simple to calculate. Since I know that 𝑎 plus 𝑏 equals four, I could subtract 𝑏 from both sides. 𝑎 in terms of 𝑏, 𝑎 equals four minus 𝑏. And in our equation, 𝑎 times 𝑏 equals three. I could substitute four minus 𝑏 for 𝑎 and then multiply both terms by 𝑏 gives us four 𝑏 minus 𝑏 squared equals three.
To set this equation equal to zero, I subtract four 𝑏 from both sides and add 𝑏 squared to both sides. Zero equals 𝑏 squared minus four 𝑏 plus three. We’ll factor this polynomial, looking for values that multiply together to equal three and add together to equal negative four. Those values would be negative one and negative three. We set each factor equal to zero. 𝑏 minus one equals zero. And 𝑏 minus three equals zero, which confirms that 𝑏 equals one and 𝑏 equals three.
