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# How to Solve Quadratic Equations

28 November,2014

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+bx+c, where a, b and c are just numbers.

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

X+ 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 x+ 5x + 6. Let us see how it works.

You can claim that

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

### Step 2

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

X+ 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,

X+ 5x +6 = x+5x +6

This is how you factor simple quadratics.

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

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