A quadratic equation is an equation that contains a squared variable, such as x squared or y squared. The usual form of a quadratic equation is Ax^2 + Bx + C. This equation says that one number times x squared plus another number times x plus another number equals zero. X is the unknown number in this equation. The x in the Ax^2 part is the same number as the x in the Bx part.
Let’s write out a quadratic equation.
2x^2 + 13x + 15 = 0
In order to solve this equation, we have to write it out in factored form. The factored form of this equation will have two expressions multiplied together on the left side instead of three numbers added together. The factored form of the equation will look like this.
(2x + ?)(x + ?) = 0
The 2x and the x come from the 2x^2 part of the equation. 2x^2 is the same thing as 2x times x or 2xx. The 2x part of 2xx goes inside the left parentheses and the other x inside of 2xx goes inside the right parentheses. Now we need to find the two missing numbers. The two missing numbers will need to multiply up to the last number on the left side of the original equation, which is 15. Let’s write out all of the pairs of numbers that multiply up to 15.
1 X 15
3 X 5
5 X 3
15 X 1
The pair of numbers will either be 1 and 15 or 3 and 5. This is how we can find the correct pair of numbers. The A number at the front of the equation next to x^2 is 2. That means that one number in the pair that we choose needs to be multiplied by 2. Once we multiply one number in the pair by 2, we have to add it to the other number in the pair. Let’s try this with the 1 and 15 pair.
The first way we could try is multiplying 1 by 2 and then adding 15. The second way we could try is multiplying 15 by 2 and then adding 1.
1(2) + 15 = 17
15(2) + 1 = 31
If we are using the correct pair, we will get 13 (the middle number from 13x) as the answer to one of them. Niether one equals 13, so 1 and 15 is not the correct pair. Let’s try 3 and 5.
3(2) + 5 = 11
5(2) + 3 = 13
5 and 3 is the correct pair. When we multiply 5 by 2 and then add 3 we get 13. We can plug in 5 and 3 where the question marks used to be. In order to get 13, we had to multiply 5 by 2. That means that 5 needs to go inside the right parentheses so that it will be multiplied by the 2x. The factored equation will look like this.
(2x + 3)(x + 5) = 0
We need to split the left side of the equation into two equations and make both of them equal zero.
2x + 3 = 0
x + 5 = 0
Now we need to solve both equations. In the first equation, we subtract 3 from both sides and we get 2x = -3. Then we divide both sides by 2 and we get x = -3/2. In the second equation, we subtract 5 from both sides and we get x = -5.
The two answers are x = -3/2 and x = -5.
Let’s try a quadratic equation with one negative number inside of it.
2x^2 – 13x + 15 = 0.
We need two numbers that multiply up to 15. The two numbers must be negative since we will be adding up to a negative number. Let’s try to use -5 and -3. When we multiply -5 by 2, we get -10. When we add -10 to -3, we get the -13 in the middle. We need to plug in -5 and -3 into the factored equation. The -5 goes inside the right parentheses since it was the number that was multiplied by something.
(2x – 3)(x – 5) = 0
Now we need to set up the two equations and make them both equal to zero.
2x – 3 = 0
x – 5 = 0
When we solve the first equation, we get x = 3/2. When we solve the second equation, we get x = 5. Those are the two answers.
Let’s try solving a quadratic equation with two negative numbers inside of it.
3x^2 -2x – 21 = 0
The 3x^2 is the same thing as 3xx, so the factored equation will have 3x inside the left parentheses and x inside the right parentheses. Now we need to find the missing pair of numbers.
The third number on the left side of the equation is -21. We need two numbers that multiply up to -21. Let’s try -3 and 7. One of these numbers needs to be multiplied by 3 since the number next to x^2 is 3. Let’s try multiplying -3 by 3. It equals -9. After we multiply -3 by 3, we need to add the other number from the pair to the answer. That means we need to add -9 + 7. It equals -2, which is the middle number on the left side of the equation. Since we got -2, -3 and 7 is the correct pair of numbers. We need to plug these numbers into the factored equation. -3 goes into the right parentheses since it was multiplied by the number next to x^2. The factored equation will look like this.
(3x + 7)(x – 3) = 0
The two equations will look like this.
3x + 7 = 0
x – 3 = 0.
When we solve the first equation, the answer is x = -7/3. When we solve the second equation, the answer is x = 3. Those are the two answers to the problem.