Solve each cubic equation using factoring and the quadratic fo...

Question

Solve each cubic equation using factoring and the quadratic formula. See Example 7.

$x^{3} + 64 = 0$

Accepted Answer

Hey, everyone here we are asked to make use of factory and the quadratic formula to solve the given cubic equation. 512 X cubed plus one is equal to zero. Now, here we are given a cubic equation. So we want to begin solving this equation by first rewriting both of our terms as cubes. So we can begin with our first term 500 and 12 X cubed, which can be rewritten as the quantity of eight X cubed. And now our second term is just one. So we have plus the quantity of one cubed and this is still equal to zero. So now that we have rewritten our expression in terms of cubed terms, we can apply the sum of cubes formula to factor this. So we need to recall the sum of cubes formula, which is A cubed plus B cubed is equal to the quantity of A plus B times the quantity of A squared minus A B plus B squared. So now we just need to identify our values for A and B according to our equation that we already have. So our first term is going to be our A term. So A according to our equation and formula is going to be A X and then that leaves our other value our other term to be B. So the value for B is just one. And now we just need to plug in these values into our formula here to find the factored form of the left hand side of our equation. So we see that the left hand side of our equation, we had the quantity of eight X cubed plus the quantity of one cubed. And now, according to our formula, we know that this expression is the equivalent to the quantity of eight X plus one times the quantity of 64 X squared minus eight X plus one. And now we can insert this factored form into our equation. And we see now that the factored form of the quantity of A X plus one times the quantity of 64 X squared minus eight X plus one is all equal to zero. And now, since these two expressions are products of each other and equal to zero, we can apply the zero property, the zero factor property which allows us to set each individual expression equal to zero. So we can begin with the first part of our expression where we have A X plus one. And again, we can set this equal to zero. And now we need to solve for X. So first, we need to subtract one from both sides and we get eight X is equal to negative one. And now to continuing to solve for X, we need to divide both sides by eight. And so we see that one solution for X is negative 1/8. And now we can do the same thing for the second part of the left side expression. So we have 64 X squared minus eight X plus one. And again, we can set this expression equal to zero. And now here we can't directly solve for X because here we have a quadratic equation. So now we need to apply the quadratic formula to find our values for X. So first, we need to recall the quadratic formula which tells us that X is equal to negative B plus or minus the square root of B squared minus four A C all over two A. And now looking at our equation that we have meaning to identify our values for A B and C. And while beginning with our value for A A is the of X squared, so A is going to be 64 then B is going to be the coefficient of X. So B is going to be negative eight. And lastly our last value, the constant is going to be the value for C. So C is just going to be one. And now we just need to plug in these values into our quadratic formula to find our other possible X solutions. So we see here that X is equal to negative quantity of negative eight plus or minus the square root of the quantity of negative eight squared minus four times 64 times one, all divided by two times 64. So now we just need to simplify this expression. So we have X is equal to positive eight plus or minus the square root of negative 192. And this is all divided by two times 64, which simplifies to 128. So now again, we need to continue to simplify. And next, we can focus on simplifying our route. So we have eight plus or minus the square root of. And now for this negative 192, we first want to factor out the negative one and then we want to rewrite 192 as a perfect square if possible. And so we can rewrite 192 as times three. So, so far underneath the route, we have negative one times 64 times three. And again, this whole expression is still divided by 128. So now we can continue to simplify the route here. And we see that X is equal to eight plus or minus. And now the square root of negative one is I, and we know the square root of 64 is eight. So this route simplifies to eight times I times the square root of three. And then again, this whole expression is still divided by 128. So now we just need to simplify our expression further. So we have eight divided by 128 which is just 1/16. So we have 1/16 plus or minus. And now we can do the same thing with the 88 I times the square root of three divided by 128 simplifies two I times the square root of three divided by 16. So now we have found two more solutions by using the quadratic formula. So we can actually rewrite our three solutions as a solution set. So we see that all of our X values we have negative 1/8 and we also have from our quadratic formula 1/16 plus or minus I times the square root of three all divided by 16. So with this, we have found three solutions for our original given cubic equation. And therefore we are left with answer choice. C Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each cubic equation using factoring and the quadratic formula. See Example 7.
$x^{3} + 64 = 0$

Key Concepts

Factoring Sum of Cubes

Solving Quadratic Equations

Zero Product Property

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