Solving Quadratic Equations :

We now move on to difficult terrain. Once you have mastered linear equation, we now must tackle quadratic equations.

A quadratic is a polynomial that looks like ax^⁰+bx+c, where a, b and c are just numbers.

In order to solve quadratic equations, you will first need to factor quadratics.

Factoring the quadratics:

For an easy case of factoring, you will need to find two numbers that will not only multiply to equal the constant term c but will all add up to equal the b the coefficient on the x term.  For instance, factor the following:

X² + 5x + 6

Step 1

You need to find factors of 6 that can add up to 5. Since 6 is the product of 2 and 3 and since 2 +3 = 5, you should take 2 and 3. B y multiplying polynomials, you can get this quadratic. So by multiplying two factors of the form “(x+m)( x+n)” where m and n are numbers, you can get the above quadratic form. So draw your parenthesis with an x in front of each number and write like this: (x+2) (x+ 3). This is how you factor the above quadratic x² + 5x + 6. Let us see how it works.

You can claim that

(x + 2) (x + 3) = X² + 5x +6

Step 2

Removing the parenthesis on the left hand side of the equation you get

X² + 3x + 2x + 6 (you simply multiply the variable and the constant of the first parenthesis with the those of the second parenthesis)

Therefore you can write,

X² + 5x +6 = x² +5x +6

This is how you factor simple quadratics.

The Quadratic formula:

Sometimes you will need to use the quadratic formula to solve quadratic equations. Remember the simple quadratic polynomial ax² + bx + c. Well, a quadratic formula is derived from the process of completing the square and is formally stated as ax² + bx + c = 0 and the value of x is given by the formula

x=(-b±√(b^2-4ac))/2a

Let us now solve a basic quadratic equation:

Solve, x² + 3x – 4 = 0

Step 1

By factoring the quadratic you know the two factors of x are -4 and 1 (because -4 + 1 = 3 and -4 x 1 = -4)

Step 2

 we substitute x with its factors. Thus, using the quadratic equation we get,

x=(-3±√(3^2-4(1)(-4))/2(1)

x=(-3±√(9+16))/2

x=(-3±√25)/2

x=(-3±5)/2

x=(-3-5)/2   and      x=(-3+5)/2

Therefore, x = -8/2 and 2/2

Or x = -4 and x = 1  

Let's watch the video demonstration of Quadratic Equations for better understanding

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