Given that four 𝑥 squared plus 24𝑥 plus 𝑐 is equal to 𝑎 times 𝑥 plus 𝑏 squared plus three 𝑏, work out the value of 𝑐.
So to find the value of 𝑐, we need to work out the right-hand side of the equation and see what 𝑐 would be equal to. Let’s first begin by expanding this 𝑥 plus 𝑏 squared. So we need to take 𝑥 plus 𝑏 times 𝑥 plus 𝑏.
This is sometimes called the FOIL method. We multiply the first numbers together, 𝑥 times 𝑥, which is where we get F. And then the next letter in FOIL is O. So we’ll multiply the outsides together, 𝑥 times 𝑏, which is 𝑏𝑥. Next is the letter I, so multiplying the insides together, 𝑏 times 𝑥, which is 𝑏𝑥, and then lastly the letter L, multiplying the last together. 𝑏 times 𝑏 is 𝑏 squared. And don’t forget to bring down the plus three 𝑏. We can combine these two middle terms. 𝑏𝑥 plus 𝑏𝑥 is two 𝑏𝑥.
Now let’s distribute the 𝑎. So we have 𝑎𝑥 squared plus two 𝑎𝑏𝑥 plus 𝑎𝑏 squared. And then bring down the three 𝑏. So right now, we have that four 𝑥 squared plus 24𝑥 plus 𝑐 is equal to 𝑎𝑥 squared plus two 𝑎𝑏𝑥 plus 𝑎𝑏 squared plus three 𝑏. So our leading terms are four 𝑥 squared on the left-hand side and 𝑎𝑥 squared on the right-hand side.
So the coefficients of 𝑥 squared have to be exactly the same. So this means that 𝑎 must equal four. So on the right-hand side of the equation, let’s replace all of the 𝑎s with four. And now that we’ve done that, we need to take two times four, which is eight. So our right-hand side of the equation is now four 𝑥 squared plus eight 𝑏𝑥 plus four 𝑏 squared plus three 𝑏. So our first terms match, and the coefficients with our 𝑥 term have to match.
So on the left-hand side of the equation, we have 24𝑥. And on the right, we have eight 𝑏𝑥. So 24 must be equal to eight 𝑏. And if we divide both sides of the equation by eight, we find that 𝑏 must equal three. So let’s replace 𝑏 with three on the right-hand side of the equation. And now we evaluate. Eight times three is 24. Three squared is nine, and then nine times four is 36. And then three times three is nine. So simplifying, we need to take 36 plus nine, which is 45. So what we have is that the four 𝑥 squareds match. The 24 𝑥s match. So this means that 𝑐 must equal 45. So our value of 𝑐 is 45.