How to solve simple Quadratic Equations(Factoring and Squaring)
Introduction
If you have seen Quadratic Equations before you know how hard and complicated it is to do with the formula. The formula is the best way but a very inefficient way in some cases. That's why in simple Quadratic Equations factoring is one of the best ways to solve them. The other way is to use squaring which is basically moving variables to one side and numbers to the others.
What are Quadratic Equations
Quadratic Equations are equations using variables, polynomials, and exponents. Quadratic equations are usually written in the format of ax² + bx + c =0. This may seem very complicated but it is not. An example of how they are written is like this, x² + 9x +15 = 0.
What is a Simple Quadratic Equation?
A simple quadratic equation is an equation in the form of ax² + bx = 0 or ax² + c =0, or in less cases ax² =0. Each equation has a different way of solving but the quadratic formula can solve all of them.
The Quadratic Formula
The quadratic formula is the formula that works on every possibility. The formula is shown down below. To use it we just substitute values in the equation.
Factoring
Factoring can solve the case of ax² + bx = 0. To use the factoring method we have to find a common factor in the equation and factor that out. Then using the zero property we can find the 2 suitable answers. For example in a question like 4x²- 3x = 0. We can see that a visible factor in 4x² and -3x is X so we can factor out x. To do so we put x on side of the equation using parentheses like this: x()=0. Then we find out what x has to be multiplied by to get 4x². We see that x*4x = 4x² so we put it into the equation and get: x(4x) =0. Next, we find what x has to be multiplied with to get -3x. We see that x * -3 = -3x so we insert that into the equation to get : x(4x-3)=0. We now have factored everything out and now we can use the zero property. The zero property is that anything times 0 is 0. So we can say that if the first x in the equation x(4x-3) = 0, is equal to 0 then the equation will be 0=0. So one of our answers is that x =0. But in quadratic equations, there are normally two answers. So to get our second answer we know that anything times 0 is 0 so if (4x-3) in the equation is equal to 0 then our equation will be true. So now we solve (4x-3) = 0 which after solving is x = 3/4. So our second answer is x = 3/4. In conclusion the answer to 4x²-3x=0 is x=0 and x = 3/4.
How to solve using Squaring
In the equation 6x²-54 = 0, we see that our format has now changed from ax² + bx =0 to ax² + c = 0. These equations can be solved using squaring. To solve it we isolate the variable and then fully isolate the variable using square roots. Our first step is to move the -54 to the right side of the equation. We can do that by adding 54 to both sides. 6x²-54+54=0+54 which is simplified to 6x²=54. Now we divide both sides by 6 to isolate the variable which is 6x²/6=54/6, which is simplified to x² = 9. Now we take the square root of both sides to remove the exponent from the left side sqrt(x²) = sqrt(9). The square root of a square is the absolute value of the number so sqrt(x²) = |x|. The square root of 9 is 3. So now our equation is |x|=3. The equation is very easy since it is direct. We have x = 3 and x =-3. X = -3 because of the absolute value sign. So in conclusion, the answers to 6x²-54=0 are x=3 and x =-3.
Simplest Quadratic Equation
The simplest version of a quadratic equation is an equation in the form of ax² =0. This form has only one answer everytime and the answer is 0. For example in the question 5x² = 0, we can solve this the normal way by dividing 5 on both sides to get 5x²/5=0/5 which is x² = 0. Then we know that 0 at the power of anything is 0 so the answer is 0 since 0² = 0. There is no other answer to this equation. Basically if you ever see a problem in the form of ax² = 0 just know that the answer is 0.
Conclusion
In conclusion when seeing problems in the form of ax² + bx =0, ax² + c, or ax² = 0, we can use factoring and squaring. These are the easiest ways to solve simple quadratic formulas and are certainly easier than using the quadratic formula. Hope that helps you!
