Solving Quadratic Equations by Taking Square Roots 07:17 minutes

Video Transcript

Transcript Solving Quadratic Equations by Taking Square Roots

Welcome back! Let’s join our game, already in progress. It’s time to play “The Difference of Two Squares”! Okay, it’s Phillip and Lara’s turn and they choose Dr. Evil's square. Okay, Dr. Evil, the question is: How do you factor x2 - 64 = 0? Dr. Evil says the answer is (x + 8)(x -8) = 0 and Phillip and Lara agree. They’re correct and get the square! And Round 1 goes to Phillip & Lara! Let’s take another look at the game-winning question.

Factoring Equations

They were asked to solve x2 - 64 = 0, which is the difference of two squares. Let’s see how they did it: First, set up your two factors like this: ( + )( - ) = 0. Then, plug square roots into the blanks. Next, complete the parentheses with the square roots of the two terms from the binomial. Plug in the square root of the first term, x², in the first place in the parentheses and the root of the second term -- disregard the minus sign -- in the second place. At last, just simplify the expressions and the equation is factored.

Now let’s join them in the bonus round already in progress. It’s Keith and Jasmine’s turn. They’re looking to come back from behind in this game. They need a block, so they pick Rapunzel. Okay, Rapunzel, your question is to factor 16x4 - 81 = 0. Rapunzel says the answer is (4x² + 9)(4x² - 9) = 0. Keith & Jasmine take a moment to talk it over and yes!!! They agree with Rapunzel. They're correct! And Keith & Jasmine win the square! What a block!
But Phillip and Lara can win the game here! Phillip and Lara pick the Dark Count’s square in hopes of ending this Bonus Round with a win. The Dark Count’s question is 49 - x⁶ = 0. Dark Count, how do you factor this one? That's a tricky one, but the Dark Count looks confident and says the answer is (7 + x³)(7 - x³) = 0. Phillip and Lara discuss the question and it looks like, yes, they do agree which is exactyl correct! Phillip and Lara win the square...and the GAME!!! Whew! that last round went quickly! Let's take a look at that again in slow motion. There were three critical points to the two teams' success.

Difference of two Squares

Let's start with a normal quadratic function: ax² + bx + c = 0. To be a difference of two squares, the 'b' coefficient must be 0 leaving us with ax² + c = 0. The coefficients 'a' and 'c' may not have the same sign. The exponent must be an even number. So let's change this plus to a minus, leaving us with ax² - c = 0. Since it's also possible that 'a' is negative and 'c' is positive, we could also see the binomial expression in the form c - ax² = 0. Keith & Jasmine had to factor a doozey of a binomial when they got 16x⁴ - 81 = 0.
But they kept their cool and set up their solution. Keith and Jasmine set up the solution the way they were taught. Now they just need to place the square root of each term of the binomial into the proper place. The square root of 16 is 4 and the square root of 'x' to the fourth is 'x' squared, making the first term for each set of parentheses 4x squared. Since the square root of 81 is 9, Keith and Jasmine write 9 in the second place of each set of parentheses and plugged in the numbers into their solution, giving them (4x² + 9)(4x² - 9) = 0.

Not to be outdone, Phillip and Lara also had a bit of a curve ball thrown their way with the expression 49 - x⁶ = 0. They went down their checklist, point by point. The 'b' coefficient must be 0, check. The coefficients 'a' and 'c' may not have the same sign, check. The exponent must be an even number, check. With all the boxes checked, they set up their solution, but with a slight twist. They use their knowledge of exponents to get their second term, +/- x³. It is just the square root of x⁶. So they plug in that number.

Zero Product Property

Then, they took the square root of 49 and plugged that answer in as well. Perfect! Remember, ladies and gents, you can solve all of these equations by simply using the Zero Product Property! Simply set each of the parentheses equal to zero and solve using PEMDAS!

Let’s join our players and crown our winner for today! Phillip and Lara seem to be enjoying a relaxing afternoon on the beach.