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### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 4

Lesson 10: Topic B: Lessons 14-15: The quadratic formula

Sal solves the equation -7q^2+2q+9=0 by using the quadratic formula. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Where do the constants come from and what do they do?
• the constants come from the "standard form" of a quadratic equation.

which is ax^2 + bx + c = 0

example: 2x^2 + 4x + 6 = 0 (so a = 2, b = 4, c = 6)
so the constants are A B and C. and if you look at a quadratic equation presented to you in standard form, you can see what A B and C are.

if youre still not sure, look at my equations again and look at a few quadratic equations and it should start to make sense

ax^2 + bx + c = 0

2x^2 + 4x + 6 = 0 (a=2,b=4,c=6)

3x^2 + 6x + 9 = 0 (a=3,b=6,c=9)

and remember if it just looks like x^2 + 4x + 8 = 0 (meaning there is nothing before the x^2 so it looks like A doesnt exist, it is there in the form of 1)

x^2 is the same as (1)(x^2) so in that case, A = 1

hope that helps!
• what if a is an irrational number? My teacher gave us a problem that like that and when i asked of that was possible she said yes. Is it?
• It sure is. Irrational numbers are just numbers, and they can be algebraically manipulated just like any others. Let's say we have the equation √5 x^2 + √3 x - π = 0. Is that a valid quadratic equation? Yes, it is. We have a = √5, b = √3, and c = -π. Plugging those into the quadratic formula gives x = [-√3 ± √((√3)^2 + 4π√5)] / 2√5. That doesn't actually simplify very much (although (√3)^2 simplifies to 3, of course), but if you use your calculator you'll see that the roots are approximately x = 0.8597 and x = -1.6343.
• Why does it have to be plus or minus?
• We used the quadratic equation because we are solving a quadratic expression. What is that? It is an expression that has a variable raised to the power 2 AND that the power 2 is the largest power in the expression: eg x²+2x+1 is a quadratic expression, but 2x+1 is not (no x²), nor is x³+x²+2x+1 (the power 3 in x³ is too big.)
OK with me so far?
Here is a simple example:
y=x² is a quadratic expression since it has a x² term and there is no other term with a greater exponent.
Now suppose I wrote 9=x². What values of x satisfy the equation?
I am sure you know that 3x3=9, so x=3 is a solution, right?
But what about -3? (-3)(-3) also equals 9, (-3)(-3)=9, so x=-3 as well. So this quadratic equation has TWO solutions.
We can write these two solutions as x=±2, which means x=2 and x=-2.

So you can see we have to use plus or minus, because when you solve a quadratic equation, very often there are two solutions, those being some value, lets call it 'a', and its negative, that is '-a'.

Keep Studying and Keep Asking Questions!
• when doing the quadractic formula would you always need a -b
• if there is nothing in front of the x (coefficient) then you take that to mean 1.
for example, x^2-x+44 is exactly the same as 1x^2-1x+44.
• When you have a quadratic equation, (i.e., 2x^2-12x-14) how would you simplify with the two negatives? I know you'd have to chose two numbers that multiply to -14 but add to -12 but how does that work? You could only get -14 and 12 instead of -12. Help!
• Halle,
For 2x^2-12x-14 your first factor out a 2
2(x^2 - 6x-7) Then you need numbers that add to -6 and multiply to -7

I hope that helps make it click for you.
• What if the radical number isn't prime? Like 16 went perfectly into 256 but what if there's a problem in which its not like that
• 2x^2 do you times it or leave it
• Since the 2x are left out of the parenthesis, it's just 2(x^2).
• This topic is generally featured in an algebra 1 course, correct?
• I would think both. I used them in Algebra 1 and 2. It should be introduced, however, during Algebra 1 and used through your math career. Even Calculus.
• Normally we solve for x but, in this one we are solving for q, can this be any letter?
But notice, Sal has `(-2-16)/(-14) = 18/14` because he cancelled out the minus in the -18 with the minus in the -14.