Content

• To step
• Method 1 of 3: Factor
• Method 2 of 3: Applying the Abc formula
• Method 3 of 3: Squared off
• Tips

A quadratic equation is an equation where the largest exponent of a variable is equal to two. Three of the most common methods of solving these equations are: factorization, use the abc formula, or split the square. If you want to know how to master these methods, just follow these steps.

To step

Method 1 of 3: Factor

1. Move all terms to one side of the equation. The first step in factoring is to move all terms to one side of the equation, keeping x positive. Apply the addition or subtraction operation to the terms x, the variable x, and the constants, moving them to one side of the equation in this way, leaving nothing on the other side. Here's how that works:
   • 2x - 8x - 4 = 3x - x =
   • 2x + x - 8x -3x - 4 = 0
   • 3x - 11x = 0
2. Factor the expression. In order to factor the expression, you have to factor out the factors of 3x, and the factors of the constant -4, in order to be able to multiply them and then add them to the value of the middle term, -11. Here's how:
   • Since 3x has a finite number of possible factors, 3x and x, you can write these in brackets: (3x +/-?) (X +/-?) = 0.
   • Then use an elimination method using the factors of 4 to find a combination that gives -11x as a result of the multiplication. You can use either a combination of 4 and 1, or 2 and 2, because the multiplication of both number combinations yields 4. Keep in mind that one of the terms must be negative, because the term is -4.
   • Try (3x +1) (x -4). When you work this out you get - 3x -12x + x -4. If you combine the terms -12x and x, you get -11x, which is the middle term you wanted to arrive at. Now you have factored this quadratic equation.
   • Another example; we try to factor an equation that does not work out: (3x-2) (x + 2) = 3x + 6x -2x -4. If you combine these terms, you get 3x -4x -4.Even though the product of -2 and 2 equals -4, the middle term doesn't work because you were looking for -11x, not -4x.
3. Determine that every pair of parentheses equals zero and treat them as separate equations. This will cause you to find two values ​​for x that both make the whole equation equal to zero. Now that you've factored the equation, all you have to do is make each pair of parentheses equal to zero. So you can write that: 3x +1 = 0 and x - 4 = 0.
4. Solve every equation. In a quadratic equation, there are two given values ​​for x. Solve each equation independently by isolating the variable and writing the results of x. Here's how to do that:
   • 3x + 1 = 0 =
   • 3x = -1 =
   • 3x / 3 = -1/3
   • x = -1/3
   • x - 4 = 0
   • x = 4
   • x = (-1/3, 4)

Method 2 of 3: Applying the Abc formula

1. Move all terms to one side of the equation and merge the like terms. Move all terms to one side of the equal sign, keeping the term x positive. Write the terms in decreasing order of magnitude, so x comes first, followed by x and then the constant. Here's how to do that:
   • 4x - 5x - 13 = x -5
   • 4x - x - 5x - 13 +5 = 0
   • 3x - 5x - 8 = 0
2. Write down the abc formula. This is: {-b +/- √ (b - 4ac)} / 2a
3. Find the values ​​of a, b, and c in the quadratic equation. The variable a is the coefficient of x, b is the coefficient of x and c is the constant. For the equation 3x -5x - 8 = 0, a = 3, b = -5, and c = -8. Write this down.
4. Substitute the values ​​of a, b, and c in the equation. Now that you know the values ​​of the three variables, you can just enter them into the equation as we show here:
   • {-b +/- √ (b - 4ac)} / 2
   • {-(-5) +/-√ ((-5) - 4(3)(-8))}/2(3) =
   • {-(-5) +/-√ ((-5) - (-96))}/2(3)
5. Calculate. After entering the numbers, you work out the problem further. Below you can read how that goes further:
   • {-(-5) +/-√ ((-5) - (-96))}/2(3) =
   • {5 +/-√(25 + 96)}/6
   • {5 +/-√(121)}/6
6. Simplify the square root. If the number under the square root is a perfect square or also a square number, then you get a whole number for square root. In other cases, simplify the square root as much as possible. If the number is negative, and you are sure that this is also the intention, then the square root of the number will be less simple. In this example, √ (121) = 11. You can then write that x = (5 +/- 11) / 6.
7. Solve for the positive and negative numbers. Once you have eliminated the square root, you can continue until you find the negative and positive answers for x. Now that you have received (5 +/- 11) / 6, you can write down the two possibilities:
   • (5 + 11)/6
   • (5 - 11)/6
8. Solve for the positive and the negative answers. Calculate further:
   • (5 + 11)/6 = 16/6
   • (5-11)/6 = -6/6
9. Simplify. To simplify, divide the answers by the largest number that is divisible for both the numerator and denominator. So divide the first fraction by 2 and the second by 6 and you have solved x.
   • 16/6 = 8/3
   • -6/6 = -1
   • x = (-1, 8/3)

Method 3 of 3: Squared off

1. Move all the terms to one side of the equation. Make sure the a of x is positive. Here's how to do that:
   • 2x - 9 = 12x =
   • 2x - 12x - 9 = 0
     • In this equation a equal to 2, b is -12, and c is -9.
2. Move the constant c to the other side. The constant is the numeric value without a variable. Move this to the right side of the equation:
   • 2x - 12x - 9 = 0
   • 2x - 12x = 9
3. Divide both sides by the coefficient of the a or x term. If x does not have a term before it and has a coefficient with the value 1, you can skip this step. In this case, you have to divide all terms by 2, like this:
   • 2x / 2 - 12x / 2 = 9/2 =
   • x - 6x = 9/2
4. Part b by two, square it and add the results to both sides of the is sign. The b in this example it is -6. Here's how to do this:
   • -6/2 = -3 =
   • (-3) = 9 =
   • x - 6x + 9 = 9/2 + 9
5. Simplify both sides. Factor the terms on the left to get (x-3) (x-3), or (x-3). Add the terms to the right to get 9/2 + 9, or 9/2 + 18/2, which adds up to 27/2.
6. Find the square root of both sides. The square root of (x-3) is simply (x-3). You can also write the square root of 27/2 as ± √ (27/2). Therefore, x - 3 = ± √ (27/2).
7. Simplify the square root and solve for x. To simplify ± √ (27/2), look for a perfect square or square number with the numbers 27 or 2 or in their factors. The square number 9 can be found in 27, because 9 x 3 = 27. To eliminate 9 from the root, write it as a separate root and simplify it to 3, the square root of 9. Let √3 be in the numerator of the fraction because it cannot be separated from 27 as a factor, and make 2 the denominator. Then move the constant 3 from the left side of the equation to the right and write down two solutions for x:
   • x = 3 + (√6) / 2
   • x = 3 - (√6) / 2)

Tips

• As you can see, the root sign has not completely disappeared. Therefore, the terms in the numerator are not merged (they are not equal terms). So it is pointless to divide the minuses and pluses. Instead, division eliminates any common factor - but "ONLY" if the factor is equal for both constants, "AND" the coefficient of the square root.