Solving Quadratic Equations Using the Square Root Property

Solving Quadratic Equations Using the Square Root Property

In this lesson we are going to cover the square root property one of the methods used to solve quadratic equations. If the quadratic equation is in standard form(vertex form) : The square root property can solve any quadratic equation written in standard form(vertex form). If given a quadratic equation in standard form(vertex form) to solve just use the square property to keep it simple.

The square root property can also solve quadratic equation in general form : The square root property is limited to the type of quadratic equations that can be solved in this form. The only equations in general form that can be solved using the square root property is if the linear coefficient(the letter b) is equal to zero.

Let's set b = 0, simplify the equation and see what happens. Adding 0 does not change the left side of the equation, and it simplifies too.

All this means is that if you want to solve a quadratic equation in general form using the square root property. The equation can only have a quadratic term and a constant term. If the equation has a linear term that is not equal to zero use another method other than the square root property to solve the equation.

Comprehension Question #1 Given the equations select all the equations that can be solved using the square root property. If the equation cannot be solved explain why not.

As we learned from an earlier lesson a quadratic equation is a second degree polynomial equation in x. The degree of a polynomial equation determines the number of solutions that you get when solving the equation. The degree(highest exponent) of a quadratic equation is 2, this means that any solving quadratic equation will result in having 2 solutions. The solutions do not have to be unique(you can get the same answer twice), but you must get 2 solutions.

Comprehension Question #2 Do not try to solve the equations. Just determine the number of solutions there will be if the equation were solved. Explain how you arrived at your answer.

Before we start solving quadratic equations using the square root property. The first example will involve solving linear equations just to review the steps. The same steps that are going to be used to solve a linear equation will be used to solve a quadratic equation in Example 2.

Example 1 Solve : Combine like terms on the left side and right side of the equation if possible. Never combine like terms on opposite sides of the = sign. We cannot combine the 2x and the x they are on opposite sides of the equation.

Neither side of the equation can be simplified further by combining like terms, the next thing is to get the "x" variable on one side of the equation. We have to options cancel out the 2x on the left side of the equation or cancel out the x on the right side of the equation. This step is always a matter of preference if done correctly we will get the same answer. In the example both options will be done.

If 2x is cancelled out. If x is cancelled out. The variable is on one side of the equation, now we have to isolate the variable. Cancel out any values that are on the same side of the equation as the variable.

Cancel out addition and subtractions, and then multiplication and division.

EXAMPLE 2 Solve : SOLUTION The equation in Example 1 was notice the only difference between Example 1 and Example 2 is that each of the x variables is being squared. Because the highest exponent is 2 this is an example of a quadratic equation.

Repeat everything that was done for Example 1. If is cancelled out. If is cancelled out.

Isolate any values that are on the same side of the equation with the variable.

How do we cancel out the 2, that is squaring the x variable? We know when solving equations if you want to cancel out a number that is mutliplying you have to divide, if you want to cancel out a subtraction then you add that number to both sides of the equation. We need to perform the inverse operation of squaring a number, which is taking the square root of that number.

Take the square root of both sides of the equation. The plus/minus symbol needs to be placed in front of the square root whenever you are solving an equation. After this example an explanation will be given to show why the plus/minus symbol is placed in front the of square root when solving a quadratic equation.

For now place just the plus/minus symbol in front of the square root. Rewrite as two equations :

Explanation of the plus/minus symbol. In the last example we had the equation : The equation simply means there is a number that when squared is equal to 16. We need to find a number that when multiplied with itself is equal to 16.

Hopefully you were able to come up with the number 4. Because is correct, but only half correct. Most of us remember the positive factors of a number, but we must not forget the negative factors. There is another number that when it is squared is equal to positive 16, and that is.

Any positive number can be obtained by multiplying two positive numbers or multiplying two negative numbers. This is the reason why we must put the plus/minus symbol in front of the square root when solving an equation. Which will result in the equation having 2 solutions as it should be when solving quadratic equations.

EXAMPLE 3 Solve : SOLUTION Combine like terms on the left side and right side of the equation if possible. Never combine like terms on opposite sides of the = sign.

The x variable is on the both sides of the equation so we need to cancel out one of them. If is cancelled out. If is cancelled out. Isolate any values that are on the same side of the equation with the variable.

+28 +28

Take the square root of both sides of the equation to cancel out the square. Do not forget to put the plus/minus symbol in front of the square root symbol. This will make sure that there are 2 solutions.

It is not possible to take the square root of a negative number and get a real number. Whenever this happens the answer will be. NO REAL SOLUTION

EXAMPLE 4 Solve : SOLUTION Simplify the expression by applying the distributive property.

Combine like terms on the left side and right side of the equation if possible. Never combine like terms on opposite sides of the = sign. The x variable is on the both sides of the equation so we need to cancel out one of them.

If is cancelled out. If is cancelled out. Isolate any values that are on the same side of the equation with the variable.

Take the square root of both sides of the equation to cancel out the square. Do not forget to put the plus/minus symbol in front of the square root symbol. This will make sure that there are 2 solutions. Switch the terms so the variable is on the left side of the equation.

Rewrite as two equations :

Comprehension Question #3 a) Explain what type of quadratic equations that are in standard form(vertex form) can be solved using the square root property. b) Explain what type of quadratic equations that are in general form can be solved using the square root property.

EXAMPLE 5 Solve : SOLUTION This is the first example of a quadratic equation in standard form(vertex form) that is being solved used the square root property. In the beginning of the notes any quadratic equation written in standard form(vertex form) should be solved by using the square root property.

NEVER distribute when you have a standard form(vertex form) of a quadratic equation. This cannot be overstated how important it is that you recognize why it is CORRECT to distribute in Example 4 but not in Example 5. With Example 4 outside of the parentheses you do not see an exponent it is okay to distribute the 3.

With Example 5 the expression inside of the parentheses is being squared. According to order of operations before we can multiply the 3 with the terms inside of the parentheses, any exponents must be simplified. This is the reason why it is NEVER okay to distribute the number outside of the parentheses.

Isolate the parentheses by cancelling any values outside of the parentheses. The "3" is multiplying the parentheses so divide to cancel it out.

Cannot isolate the x variable because the parentheses is being squared so cancel out the exponent before the +7 can be cancelled out. Take the square root of both sides of the equation to cancel out the square. Do not forget to put the plus/minus symbol in front of the square root symbol. This will make sure that there are 2 solutions.

We are still not done because the variable is not isolated. There are 2 choices once you get to this step, once again it is a matter of personal preference.

Before we continue we must understand what the equation means. This equation states that "x + 7" is equal to positive 6 and at it also states "x + 7" is equal to negative 6. Now we can write 2 separate equations. x + 7 = 6 x + 7 = 6

Solve each equation to come up with the 2 solutions. x + 7 = 6

EXAMPLE 6 Solve : SOLUTION NEVER distribute when you have a standard form(vertex form) of a quadratic equation. Isolate the parentheses by cancelling any values outside of the parentheses.

+18 +18

Now we can write 2 separate equations.

Solve each equation to come up with the 2 solutions.

Comprehension Question #4 Given a quadratic equation : list the terms in the order that they must be cancelled out to correctly solve equation. It is not necessary to solve the equation just what must be done in that step. The first step has been completed state the remaining steps. Step 1 : Subract 11 from both sides of the equation.