Author:
Laura McKinney
Date Of Creation:
8 August 2021
Update Date:
1 July 2024
Content
A quadratic equation is a one-variable polynomial where 2 is the highest exponent of that variable. There are three main ways to solve quadratic equations: 1) factor the equation into factors if possible, 2) use the quadratic formula, or 3) complete the square. Follow these steps to learn how to become proficient with these three methods.
Steps
Method 1 of 3: Analysis of equations into factors
Add all the same terms and move them to one side of the equation. The first step in factor analysis is to put all of its terms to the side so that they are positive. To combine terms, add or subtract all terms, any containing terms, and constants (the terms are integers), convert them to one side, and leave nothing on the other side. You can then write "0" on the other side of the equal sign. Here's how to do it:
Analyze the expression into the factor. To factor an expression, you must use the factors of the term containing (3) and the factors of the constant (-4), to multiply them and then add it to the center term (-11). . Here's how to do it:- Since there is only one possible factor set, and, you can rewrite it in parentheses like this:.
- Next, use reduction to combine the factors of 4 to find the combination that makes -11x when multiplied. You can use 4 and 1 or 2 and 2 because they both have a product of 4. Just remember that a factor must be negative because our term is -4.
- With the test method, we will check the combination of factors. When we implement multiplication, we obtain. Add up the terms and, we have, is the exact middle term we're aiming for. So we have just factorized the quadratic function.
- As an example of this test, let's examine a faulty (incorrect) combination of: =. Combining these terms, we will obtain. Although it is true that -2 and 2 have products equal to -4, the term in between is not correct, because we need it, not.
Let each expression in parentheses be zero as individual equations. From there, find two values of that make the overall equation equal to zero = 0. Now, once you factor the equation, you just need to enclose the expression in parentheses with zero. Why? That's because for zero product, we have a "principle, law or property" that a factor must be zero. Therefore, at least one value in parentheses, must be zero; that is (3x + 1) or (x - 4) must be zero. So we have either.
Solve each of these "zero" equations independently. The quadratic equation has two possible solutions. Find each possible solution for the variable x by separating the variable and writing down its two solutions as the final result. Here's how:- Solve 3x + 1 = 0
- Subtract two sides: 3x = -1 .....
- Split the sides: 3x / 3 = -1/3 .....
- Collapse: x = -1/3 .....
- Solve x - 4 = 0
- Subtract two sides: x = 4 .....
- Write your own possible solutions: x = (-1/3, 4) ....., that is, x = -1/3, or x = 4 are both correct.
- Solve 3x + 1 = 0
Check x = -1/3 in (3x + 1) (x - 4) = 0:
Instead of an expression, we have (3 + 1)( – 4) ?=? 0..... Collapse: (-1 + 1) (- 4 1/3)? =? 0 ..... Perform multiplication, we get (0) (- 4 1/3) = 0 ..... 0 = 0 ..... Right, x = -1/3 is a solution of equation.
Check x = 4 in (3x + 1) (x - 4) = 0:
Instead of an expression, we have (3 + 1)( – 4) ?=? 0 ..... Collapse, we get: (13) (4 - 4)? =? 0 ..... Perform multiplication: (13) (0) = 0 ..... 0 = 0 ..... Right, x = 4 is a solution of the equation.- So both of these possible solutions have been "tested" individually, and it can be confirmed that both solve the problem and are two separate true solutions.
Method 2 of 3: Use the quadratic formula
Add all the same terms and move them to one side of the equation. Moves all terms to one side of the equal sign so that the term contains the positive sign. Rewrite the terms in descending order, meaning that the term comes first, followed by, and finally the constant. Here's how:
- 4x - 5x - 13 = x -5
- 4x - x - 5x - 13 +5 = 0
- 3x - 5x - 8 = 0
Write down your quadratic formula. That is:
Determine the values of a, b, and c in the quadratic equation. Out a is the coefficient of x, b is the coefficient of x and c is a constant. With the equation 3x -5x - 8 = 0, a = 3, b = -5, and c = -8. Please write down on paper.
Plug the values of a, b, and c into the equation. Now that you know the values of the three variables above, you can put them into the equation like this:
- {-b +/- √ (b - 4ac)} / 2
- {-(-5) +/-√ ((-5) - 4(3)(-8))}/2(3) =
- {-(-5) +/-√ ((-5) - (-96))}/2(3)
Perform calculations. After you've replaced the numbers, perform the rest of the calculation to reduce the positive or negative signs, multiply or square the remaining terms. Here's how:
- {-(-5) +/-√ ((-5) - (-96))}/2(3) =
- {5 +/-√(25 + 96)}/6
- {5 +/-√(121)}/6
Collapse the square root. If under the radical sign is a perfect square, you will get an integer. If it is not a perfect square, then reduce it to its simplest radical form. If it's negative, and you're sure it should be negative, the solution will be quite complicated. In this example, √ (121) = 11. We could write: x = (5 +/- 11) / 6.
Solve for the positive and negative solutions. If you've removed the square root, you can continue until you find the positive and negative solutions of x. Now that you have (5 +/- 11) / 6, you can write two options:
- (5 + 11)/6
- (5 - 11)/6
Find the positive and negative solutions. We just have to do the calculation:
- (5 + 11)/6 = 16/6
- (5-11)/6 = -6/6
Collapse. To shorten your answers, you just need to divide both the numerator and the model by their greatest common divisor. Divide the numerator and denominator of the first fraction by 2 and the denominator and the denominator of the second fraction by 6, and you have found x.
- 16/6 = 8/3
- -6/6 = -1
- x = (-1, 8/3)
Method 3 of 3: Complete the square
Move all terms to one side of the equation. Make sure that a or x has a positive sign. Here's how:
- 2x - 9 = 12x =
- 2x - 12x - 9 = 0
- In this equation, a equal 2, b equal to -12 and c equal to -9.
Moved on c or constant to the other side. Constants are numerical terms that contain no variables. Let's move it to the right side of the equation:
- 2x - 12x - 9 = 0
- 2x - 12x = 9
Divide both sides by the coefficients a or the coefficient of x. If x has no term in front, then its coefficient is 1 and you can skip this step. In our case, you would have to divide all terms in the equation by 2, like this:
- 2x / 2 - 12x / 2 = 9/2 =
- x - 6x = 9/2
Share b by two, square it and add the result to both sides. In this example, b equals -6. We do the following:
- -6/2 = -3 =
- (-3) = 9 =
- x - 6x + 9 = 9/2 + 9
Collapse two sides. To factor the left-hand side, we have (x-3) (x-3), or (x-3). Add the right side to get 9/2 + 9, or 9/2 + 18/2, and get 2/27.
Find the square root of both sides. The square root of (x-3) is (x-3). You can express the square root of 27/2 as ± √ (27/2). So, x - 3 = ± √ (27/2).
Collapse the radical sign and find x. To reduce ± √ (27/2), we find a square within 27, 2 or a factor of it. The perfect square 9 is in 27, because 9x3 = 27. To remove 9 from the radical sign, we pull it out and write 3, its square root, in addition to the radical sign. The remaining factor of 3 in the numerator cannot be outputted, so it remains below the radical sign. At the same time, we also leave 2 in the sample of the fraction. Next, move the constant 3 on the left side of the equation to the right, and write down the two solutions:
- x = 3 + (√6) / 2
- x = 3 - (√6) / 2)
Advice
- As can be seen, the radical sign does not disappear completely. Therefore, terms in the numerator cannot be cumulative (because they are not terms of the same property). Therefore, the plus-or-minus division is meaningless. Instead, we can divide all the common factors but JUST when constant AND Coefficients of any radical also contain that factor.
- If the radical sign is not a perfect square, the last few steps may be taken slightly differently. Such as:
- If "b" is an even number, the formula will be: {- (b / 2) +/- √ (b / 2) -ac} / a.
