Video: Solving Quartic Equations

Solve the equation (3𝑥 − 1)(5𝑥 + 6)(3𝑥 − 4)(8𝑥 + 7) = 0.

02:19

Video Transcript

Solve the equation three 𝑥 minus one multiplied by five 𝑥 plus six multiplied by three 𝑥 minus four multiplied by eight 𝑥 plus seven is equal to zero.

As you can see here, in this equation, we actually have four parentheses. So we’ve got four parentheses here. And we can actually say if any of the parentheses are equal to zero, then the solution of the whole equation will be equal to zero because it doesn’t matter what the other parentheses equal. If one of them equals zero because it’s being multiplied, then the whole equation will be equal to zero.

So therefore, to solve this equation, what I’m gonna do is actually to set up four equations. And I’m actually going to make each of the parentheses equal to zero as this will allow me to find all the possible solutions to our equation. And we start with the top left equation. So first of all, I’m going to add one to each side. So I’m left with three 𝑥 is equal to one. And then we divide both sides by three. So it gives us a final answer, which is 𝑥 is equal to a third.

So here we go. We’ve solved our first parenthesis. And now, we’re gonna move onto the next one, which is the bottom left. So we’ve got five 𝑥 plus six is equal to zero. We’re gonna start by subtracting six from each side. That gives us five 𝑥 is equal to negative six. And then we divide both sides by five to give us our answer here, which is 𝑥 is equal to negative six over five. So great! Two parentheses down, two to go.

So at the top right equation, we’ve got three 𝑥 minus four is equal to zero. I’ve added four to both sides. So I’m left with three 𝑥 is equal to four. And then if I divide three by three, I get 𝑥 is equal to four over three. So great! That’s three of our parentheses solved.

Now, the final one is the bottom right. We’ve got eight 𝑥 plus seven is equal to zero. Well, if I subtract seven from each side, I get eight 𝑥 is equal to negative seven. Then, I divide three by eight. And I get my final solution 𝑥 is equal to negative seven over eight.

So therefore, we can say that the solutions to our equation are 𝑥 is equal to a third, 𝑥 is equal to negative six over five, 𝑥 is equal to four-thirds, and 𝑥 is equal to negative seven over eight.

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