M103

Quadratic Formula-1

MathematicsGrade 12Grade 11Grade 10Grade 9

Quadratic Equations and Inequalities

Definition:

The "Standard Form" of a Quadratic Equation is:

ax2 + bx + c = 0

x represents an unknown, and “a”, “b”, and “c” represent known numbers such that “a” is not equal to 0.

A Quadratic Inequality is just like a quadratic equation, except instead of an equal sign there's an inequality.

ax2 + bx + c < 0

Everyday Use:

Figuring Out a Profit
Finding a Speed
Calculating Areas of a Structure

How to Solve Quadratic Equations:

You are able to solve Quadratic equations in these ways:

By Factoring
And Quadratic Formula

Factoring:

x2 + 5x + 6

x2 + 5x + 6 = (x+2)(x+3)

Now set the factors equal to 0.

(x+2)(x+3) = 0

Solve each factor.

x + 2 = 0 or x + 3 = 0 ?

x = –2 or x = – 3

So x = -3, -2

Quadratic Formula:

The Quadratic Formula is:

x = -b ±√b² - 4ac

Lyrics

Quadratic Formula

Hey Kids, Professor Eggbeard here

In today’s lesson I am going to tell you about the quadratic formula yes

First thing we are going to write is x= -b

On the right of the - b we are going to put a + sign

And under the + sign we are going to put a –

Now to the right of that we are going to make a radical sign

Inside of the radical sign we are going to put

B2 – 4ac

Next we are going to divide it by the coefficient 2a under the equation

And that is the quadratic formula

Now, let’s do one equation so you will understand it

Right now 2x^2 + 2x -5

Now to begin it has to be in standard form so make sure it equals zero

Let’s begin

The coefficient for 2x^2 is 2 and that’s going to be our A in the formula

Plus 2x the coefficient for 2x is 2

And that’s going to be our b

Leaving our c to be negative

Now let’s plug in our numbers, yes?

X = -2 +- the square root of 2 squared yes

Minus 4 times A which is 2 times c which is negative 5

Divided by 2a

Which means 2x2 and all we did was plug in our numbers

Now the only thing left is to work everything out

By order of operations

So let’s work out what’s in the radical sign first

2^2 is 4

Now, don’t subtract that’s considered negative

So negative 4x2 which is -8x-5

Which then is positive 40

So now you have 4+40 which then is 44

Because on the bottom 2x2= 4

So 44 over 4

Now, the square root of 4 can also be written as

A square root of 4 times a square root of 11

Standards

CCSS.MATH.CONTENT.HSA.REI.B.4.A — Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
CCSS.MATH.CONTENT.HSA.REI.B.4 — Solve quadratic equations in one variable.
CCSS.MATH.CONTENT.HSN.CN.C.7 — Solve quadratic equations with real coefficients that have complex solutions.
CCSS.MATH.CONTENT.HSA.REI.B.4.B — Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.