Math Help

Archive for the ‘Quadratic Equations’ Category

Solving Quadratic Equations (continued…)

Step 2: Equate your two simultaneous equations.

Since both equations are equal to y, we just cut off the middle man (y) and equate the two equations. The left hand side is one equation and the right hand side is the other as shown below. It does not matter which side is which.

2x ² + 6x – 6 = 14x – 12

Step 3: Grouping terms to the left hand side.

Next, you want to group all the x ² terms together, the x terms together, and the number terms together. In the above equation, there is only one x ² term so no grouping is needed. There are, however, two terms involving x (6x and 14x). We want to group them all onto the left hand side of the equation so we bring 14x from the tight hand side to the left hand side. And, we do the same thing with 12.

This is as shown:

2x ² + 6x – 6 = 14x – 12

2x ² + 6x – 6 – 14x + 12 = 0

2x ² – 8x + 6 = 0

Step 4: Solve the quadratic equation.

Since the above quadratic equation (and all quadratic equations) are of the form ax ² + bx + c = 0, we can use the Quadratic Formula. Note that if you can see an easier way to spot the quadratic solutions, then there is no need to use the Quadratic Formula. The Quadratic Formula is only the back up when you cannot easily spot the quadratic roots by other methods.

In this case a = 2, b = -8, and c = 6

The root of x is then 3 or 1.

Since y = 14x – 12, y = 30 or 2.

Quadratic equations are not difficult to solve if you follow a process. Always use the same logic to solve quadratic equations. The method of solving quadratic equations below will show you how to solve quadratic equations every time. Whenever you are faced with solving quadratic equations, you can use the following method of solving quadratic equations. 

Quadratic equation and simultaneous equations you have to solve

Suppose you are trying to solve the following quadratic equation simultaneous equations: 

y = 2x ² + 6x – 6 

y + 12 = 14x 

Solution to the simultaneous equations involving quadratic equation

Step 1: Rearrange the simultaneous equations 

The first thing to do in solving simultaneous equations involving quadratic equations is to re-organize the equations so that we can compare apples to apples. In another word, re arrange the simultaneous equations so that y is on the left hand side for both equations on its own as shown below. 

• The first equation is y = 2x ² + 6x – 6. Since y is already on the right hand side on its own, there is nothing to be done for the first equation at this stage.

Now we have two simultaneous equations involving quadratic equation as follows: 

y = 2x ² + 6x – 6 

y = 14x � 12

This section of Math Help online shows you the proof of Quadratic Formula. The Quadratic Formula is very important and should be learned. 

The proof of Quadratic Formula

If ax ² + bx + c = 0 

Then diving the above quadratic equation by a gives: 

Re arranging the above quadratic equation gives: 

Here comes the neat trick to solve this quadratic equation, add b 2 / 4a 2 to both sides of the quadratic equation in order to complete square on the left hand side. This gives the resulting quadratic equation: 

Then factorizing the left hand side of the quadratic equation gives: 

Re-arranging the right hand side of the quadratic equation gives: 

Then it is a matter of taking square root of the right hand side and rearranging the quadratic equation as shown below: