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

Question

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

$x^{3} - 27 = 0$

Accepted Answer

Hey, everyone here, we are asked to make use of factoring and the quadratic formula to solve the given cubic equation. Eight X cubed minus 343 is equal to zero. Now, to begin solving this equation, we first want to rewrite our cubic equation as cubed terms. So we can begin with our first term eight X cubed and rewrite as the quantity of two X cubed. And then we have minus 343 which can be written as minus the quantity of seven cubed. And this whole expression is still equal to zero. And now we want to factor the left side of our equation by using the difference of cubes formula. And if we recall the difference of cubes formula, we have a cubed minus being cubed is equal to the quantity of A minus B times the quantity of A squared plus A times B plus B squared. And now we just need to identify our values for A and B according to our equation that we have. So our, our first term is going to be our value for A. So A, in our case is two X and that leaves the other value to be the value for B. So B is going to be seven. And now we can just substitute in these values into the formula here to find our factored form of the left side of our given equation. So originally, we have the quantity of two X cubed minus the quantity of seven cubed. And now, according to our formula, we know that the factored form of this expression is the quantity of two X minus seven times the quantity of four X squared plus 14 X plus 49. And now we can substitute in this factored form of our expression into the left side of our given equation. So we have that the quantity of two X minus seven times the quantity of four X squared plus 14 X plus 49 is all equal to zero. And now, since we have the product of two expressions here set equal to zero, we can apply the zero factor property which allows us to set each individual expression equal to zero. So we can start with the first part of the left side of our equation. So we have two X minus seven. And again, we can set this expression equal to zero. And now we just need to solve for X. So first, we need to add seven to both sides and we get two X is equal to seven. And now to get rid of the coefficient, we need to divide both sides by two. And we see that one of our solutions for X is seven halves. And so now we can repeat this with our other part of our expression, which we have four X squared plus 14 X plus 49. And again, we can set this expression equal to zero. And now here we can't directly solve for X, but we can apply the quadratic formula to find our X values because here we have a quadratic equation. 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 C squared minus four A C all divided by two A. So now we just need to identify our values for A B and C according to our equation that we have. So starting with our value for A A is going to be the coefficient of X squared. So A in our case is four and then because coefficient of X is our value for B, so B is going to be 14 and then the value of our constant is going to be the value for C. So C is 49. And now we just need to substitute in these values into our quadratic formula to find values for X. So we see that X is equal to negative plus or minus the square root of squared minus four times four times 49 all divided by two times four. And now we just need to simplify this expression and to find a simplified value of X. So we have negative 14, we have negative 14 plus or minus the square root of. And our, our root simplifies to negative 588. And this is all divided by two times four, which is just eight. So now continuing to simplify, we can next focus on just simplifying the route. So we have negative 14 plus or minus. And for our routine, We just want to rewrite the term underneath it. So we have the square root of and now we want to first factor out our negative sign here in our negative 588. So we factor out negative one and then we want to rewrite 588 as as many perfect squares as possible. So 588 can also be written as 196 times three. So again, underneath the route, we now have negative one times 196 times three. And this whole expression is still divided by eight. So now we just need to continue to simplify further and we have negative 14 plus or minus. And now the square root of negative one is I in the square root of 196 is just 14. So we have plus or minus 14, I times the square root of three. And this is all divided by eight. So now we just need to continue to simplify further here. So we have negative 14 divided by eight, which is simply divide as negative 7/ plus or minus. And we can perform the same simplification to the other part of our solution here. So we have plus or minus seven, I times the square root of three all divided by four. So with this, we have found two more solutions because of this plus or minus from our quadratic formula. So we can rewrite our total of three solutions as a solution set. And we see that the values the solutions for X R seven halves as well as negative 7/4 plus or minus seven times I times the square root of three all divided by four. So with this, we have found all of our solutions for our given cubic equation. So this is our final answer which is answer choice. D 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.
$x^{3} - 27 = 0$

Key Concepts

Factoring Cubic Equations

Difference of Cubes Formula

Quadratic Formula

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