Lecture: Quadratic Equations
Lecture script · about 15–20 minutes spoken · Week 7
Opening
A sports analyst wants to know when a kicked ball comes back down. A landscaper needs a patio that covers a set area. A shop owner wants the price where a profit curve hits zero. All three solve one kind of equation, where the unknown number is squared. Once you can solve one and check the answer, you can also tell which answers make sense in the real setting.
Here's where we are in this course. In Week 1 you learned that a negative number squared is positive. In Week 2 you solved linear equations by doing the same thing to both sides. In Weeks 3 to 5 you built functions, lines, and systems. In Week 6 you multiplied and factored polynomials, so x squared minus 6x plus 5 became x minus 1 times x minus 5. In this lecture we set that product equal to zero and solve. Next week, in Week 8, you review Weeks 1 to 7 and take the midterm.
I'll take eight ideas in order: standard form, factoring, a trap about dividing, the square root property, completing the square, the quadratic formula, the discriminant, and choosing a method. Most examples come from one equation, x squared minus 6x plus 5 equals 0, with a launched object and a patio along the way.
1. Spot a quadratic equation: standard form
Picture a square rug. If each side is x feet, the area is x times x, which we write as x squared. Any equation with a squared unknown, and no higher power, belongs to this week.
A quadratic equation is an equation in one variable that can be written as a x squared plus b x plus c equals 0, with a not equal to zero, so the highest power of the variable is 2. The neat arrangement has a name. The standard form of a quadratic equation is that arrangement with every term on the left side and 0 on the right, so the coefficients a, b, and c can be read off with their signs.
Let's read the week's main equation, x squared minus 6x plus 5 equals 0. The right side is already zero. No number is written in front of x squared, so a is 1. The x term is minus 6x, and the minus sign belongs to b, so b is negative 6. The number standing alone is 5, so c is 5.
Sometimes you have to rearrange first. The equation x squared equals 7x isn't in standard form yet. Subtract 7x from both sides, and you get x squared minus 7x equals zero. Now a is 1, b is negative 7, and c is zero.
Pause and try this. In x squared minus 6x plus 5 equals 0, is b equal to 6 or to negative 6?
Say your answer out loud before you read on.
Here's the answer. b is negative 6, because a coefficient carries the sign written in front of it, and that sign will matter in the formula later.
2. Factor and use the zero-product property
Here's a fact you've used without naming it. If two numbers multiply to zero, at least one of them must be zero.
That fact has a name. The zero-product property is the rule that if a product of factors equals 0, then at least one of the factors equals 0, so each factor can be set equal to 0 and solved.
Let's use it on x squared minus 5x plus 6 equals 0. From Week 6, you need two numbers whose product is 6 and whose sum is negative 5. Negative 2 and negative 3 work. So the equation becomes x minus 2 times x minus 3 equals zero. Now split it: x minus 2 equals zero, or x minus 3 equals zero. So x is 2 or x is 3. Check 2: 4 minus 10 plus 6 is zero. Check 3: 9 minus 15 plus 6 is zero. Both work.
The week's main equation goes the same way. You need a product of 5 and a sum of negative 6, which is negative 1 and negative 5. So x minus 1 times x minus 5 equals zero, and x is 1 or 5. Check 1: 1 minus 6 plus 5 is zero. Check 5: 25 minus 30 plus 5 is zero. If you graph the curve, it crosses the horizontal axis at exactly those two inputs.
Pause and try this. The factor x minus 2 equals zero. Does that give x equals 2 or x equals negative 2?
Think about it for a moment, then keep going.
Here's the answer. x equals 2, because 2 minus 2 is zero, while negative 2 minus 2 is negative 4.
3. Never divide by the variable
Imagine a friend looks at x squared equals 7x and says, "Divide both sides by x, and x is 7." That feels efficient. But try zero in the original equation. Zero squared is zero, and 7 times zero is zero. So zero is an answer too, and the division threw it away.
Here's why. Dividing by x quietly assumes x isn't zero. In Week 2 you divided by numbers you knew weren't zero. In a solve, you don't know that about x yet. So the safe move is factoring, not dividing.
Take 3x squared minus 12x equals 0. Both terms share 3x, so you factor it out: 3x times x minus 4 equals zero. Then 3x equals zero gives x equals zero, and x minus 4 equals zero gives x equals 4. Check 4: 3 times 16 is 48, and 48 minus 48 is zero. If you had divided by 3x, only the 4 would have survived.
The same move rescues x squared equals 7x: factor it as x times x minus 7, so x is zero or 7.
Pause and try this. A classmate solves 3x squared minus 12x equals 0 by dividing both sides by 3x and gets only x equals 4. Which answer is missing?
Take a moment, then read on.
Here's the answer. x equals zero is missing, because 3 times zero squared minus 12 times zero is zero, and dividing by 3x threw that answer away.
4. Undo a square: the square root property
Picture a square rug that covers 49 square feet. How long is a side? You need a number that times itself is 49, and 7 works. But negative 7 times negative 7 is also 49. A rug can't have a negative side, but a bare equation doesn't know about rugs.
That's the next tool. The square root property is the rule that if a square equals a nonnegative number k, then the quantity being squared equals the positive or the negative square root of k, which gives two answers when k is positive. We write the pair with the plus-or-minus sign, the symbol placed between two numbers to name both their sum and their difference at once. So x squared equals 49 gives x equals plus or minus 7: x is 7 or negative 7.
The property works on a shifted square too. Take x minus 3, all squared, equals 16. The thing being squared is x minus 3. So x minus 3 is plus or minus 4. Then x is 3 plus 4, which is 7, or 3 minus 4, which is negative 1. Check negative 1: negative 1 minus 3 is negative 4, and negative 4 squared is 16. Both answers sit 4 steps from 3, one on each side.
Pause and try this. Your calculator's square root key gives 7 for the square root of 49. Does x squared equals 49 have one solution or two?
Decide, then keep reading.
Here's the answer. It has two, 7 and negative 7, because negative 7 squared is also 49; the calculator key gives only the principal, nonnegative root.
5. Build a square: completing the square
The square root property is fast, but only when one side is already a square. Most equations aren't. So let's build one. Picture tiles: one big x by x square, some long strips, and a few small unit tiles. Split the strips evenly along two sides of the big square, and a small corner is left empty. Fill that corner, and you've completed the square.
Completing the square is a solving method that adds the square of half the x-coefficient to both sides of x squared plus b x equals a number, turning the left side into a perfect square.
Here it is on the week's main equation. Start with x squared minus 6x plus 5 equals 0. Subtract 5 from both sides: x squared minus 6x equals negative 5. Half of negative 6 is negative 3. Negative 3 squared is 9. Add 9 to both sides. The left side becomes x minus 3, all squared, and the right side becomes negative 5 plus 9, which is 4. So x minus 3, squared, equals 4. Now the square root property finishes it: x minus 3 is plus or minus 2, so x is 5 or 1.
Notice what happened. Those are the same two answers factoring gave. Two methods agreeing is a check in itself.
The one slip to watch: the 9 goes on both sides. An equation is a balance, and adding to one side alone tips it.
Pause and try this. Completing the square on x squared plus 6x plus 5 equals 0, what number do you add to both sides?
Work it out before you read on.
Here's the answer. You add 9, because half of 6 is 3 and 3 squared is 9; then x plus 3, squared, equals 4, so x is negative 1 or negative 5.
6. Use the quadratic formula
If you complete the square once on a x squared plus b x plus c equals 0, with letters instead of numbers, you get a formula that does the job every time. The quadratic formula gives every real solution from the three coefficients. In words: x equals negative b, plus or minus the square root of b squared minus 4 a c, all divided by 2a.
Let's run it on the week's main equation. a is 1, b is negative 6, and c is 5. The part under the root is negative 6 squared minus 4 times 1 times 5, which is 36 minus 20, or 16. The square root of 16 is 4. Negative b is positive 6. So x is 6 plus or minus 4, all over 2. That's 10 over 2, which is 5, or 2 over 2, which is 1. One equation, three methods, one answer.
The formula shines when factoring fails. Take x squared minus 4x plus 1 equals 0. Here b is negative 4, so negative b is positive 4. The part under the root is 16 minus 4, which is 12. The square root of 12 is 2 times the square root of 3. So x is 4 plus or minus 2 root 3, all over 2. Divide both terms on top by 2, and x is 2 plus or minus the square root of 3. As rounded decimals, that's about 3.732 and about 0.268. The exact answer keeps the radical; the decimals check the size.
Two slips live here. Some people write negative b as negative 4 when b is negative 4. Others divide only the radical by 2. The whole top sits over 2a.
Pause and try this. Someone solves x squared minus 4x plus 1 equals 0 and writes 4 plus or minus the square root of 3. Which slip did they make?
Find the slip before you read on.
Here's the answer. They divided only the radical by 2; the whole top, 4 plus or minus 2 root 3, has to be divided by 2, which gives 2 plus or minus root 3.
7. Read the discriminant before you solve
Look again at the formula. Everything hangs on the number under the square root. If it's positive, plus and minus give two different answers. If it's zero, adding and subtracting zero lands on one number. If it's negative, no real number has that square root.
That number is the discriminant: the quantity b squared minus 4 a c, the part under the square root in the quadratic formula. Its sign tells you before you solve whether a quadratic equation has two real solutions, one repeated solution, or no real solution.
For the week's main equation it's 16, a positive number, so two real solutions, 1 and 5. Now take 2x squared minus 4x plus 2 equals 0. The discriminant is 16 minus 16, which is zero. That gives a repeated solution, the single answer you get when plus and minus land on the same number. Here x is 4 over 4, which is 1. Last, take x squared plus 2x plus 5 equals 0. The discriminant is 4 minus 20, which is negative 16. That means no real solution: no real number makes the equation true. Mathematicians do have a larger system, called the complex numbers, where this equation has solutions, but that system is outside this course.
Pause and try this. An equation's discriminant is negative 16. Someone writes x equals negative 4. What would you tell them?
Think it through, then read on.
Here's the answer. Negative 4 squared is positive 16, not negative 16, so no real number works and the right response is the words no real solution.
8. Apply the methods and choose one on purpose
Back to the launched object. It leaves an 80-foot platform going up at 64 feet per second, and its height after t seconds is negative 16 t squared plus 64t plus 80 feet. That 16 is half of gravity's pull of 32 feet per second per second. Its heights at 0 through 5 seconds run 80, 128, 144, 128, 80, and 0 feet.
When does it land? Set the height to zero. Divide every term by negative 16, and you get t squared minus 4t minus 5 equals zero. Factor: t minus 5 times t plus 1. So t is 5 or negative 1. Now ask the setting. Negative 1 second is before the launch, so you reject it. The object lands at 5 seconds. Check: negative 400 plus 320 plus 80 is zero.
The patio works the same way. It's 3 feet longer than it is wide and covers 40 square feet. Call the width w. Then w times w plus 3 equals 40, which becomes w squared plus 3w minus 40 equals zero. That factors as w plus 8 times w minus 5, so w is negative 8 or 5. A width can't be negative, so the patio is 5 feet wide and 8 feet long, and 5 times 8 is 40.
So how do you choose? If the numbers factor at a glance, factor. If one square sits alone on one side, use the square root property. If nothing factors, use the formula. Completing the square always works, and it's how the formula was built.
Pause and try this. Which method would you reach for first on x minus 3, all squared, equals 16, and which on x squared minus 4x plus 1 equals 0?
Pick both before you read on.
Here's the answer. The square root property for the first, since one square is set equal to a number, and the formula for the second, since it has no whole-number factors.
Closing
Let's pull it together, in the order we took it. A quadratic equation goes into standard form first, with zero on one side, and each coefficient keeps its sign. Factoring plus the zero-product property turns one hard equation into two easy ones. Never divide by the variable; factor it out instead. A square equal to a number gives a plus-or-minus pair. Completing the square makes a square on purpose, and the quadratic formula does that work in one step for any quadratic. The discriminant tells you two, one, or no real solutions before you start. And every answer gets checked, then judged against the setting.
Next in this module, you'll find the slide deck, then the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Week 8, Midterm Review & Exam, you practice choosing a method for every topic from Weeks 1 to 7, and then you take the midterm.
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