3 Ways to Solve Quadratic Equations

Table of contents:

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

A quadratic equation is a mathematical equation in which the highest power of x (degree of the equation) is two. Here is an example of such an equation: 4x² + 5x + 3 = x² - 5. Solving this type of equation is complicated, since the methods used for x² they don't work for x, and vice versa. Factoring the quadratic term or the use of the quadratic formula are two methods that help solve a second degree equation.

Steps

Method 1 of 3: Using factoring

Step 1. Write all terms on one side, preferably on the side where x² it's positive.

Step 2. Factor the expression

Step 3. In separate equations, equal each factor to zero

Step 4. Solve each equation independently

It would be better not to write the improper fractions as mixed numbers, even if it would be correct from a mathematical point of view.

Method 2 of 3: Using the quadratic formula

Write all terms on one side, preferably on the side where x² it's positive.

Find the values of a, b and c. a is the coefficient of x², b is the coefficient of x and c the constant (it does not have an x). Remember to also write the sign of the coefficient.

Step 1. Find the product of 4, a and c

You will understand the reason for this step later.

Step 2. Write the quadratic formula, which is:

Step 3. Substitute the values of a, b, c, and 4 ac into the formula:

Step 4. Adjust the numerator signs, finish multiplying the denominator and calculate b ².

Note that even when b is negative, b² it's positive.

Step 5. Finish the part under the square root

This part of the formula is called "discriminant". Sometimes it's best to calculate it first, as it can tell you in advance what kind of result the formula will give.

Step 6. Simplify the square root

If the number under the root is a perfect square, you will get an integer. Otherwise, simplify down to the simplest quadratic version. If the number is negative, and you are sure it should be negative, then the root will be complex.

Step 7. Separate the plus or minus into plus option or minus option

(This step only applies if the square root has been simplified.)

Step 8. Calculate the plus or minus possibility separately

Step 9

.. and reduce each to a minimum.

Improper fractions don't have to be written as mixed numbers, but you can do it if you want.

Method 3 of 3: Complete the square

This method may be easier to apply with a different type of quadratic equation.

Ex: 2x² - 12x - 9 = 0

Step 1. Write all terms on one side, preferably on the side where a or x² are positive.

2x² - 9 = 12x2x² - 12x - 9 = 0

Step 2. Move c, or constant, to the other side

2x² - 12x = 9

Step 3. If necessary, divide both sides by the coefficient of a or x².

x² - 6x = 9/2

Step 4. Divide b by two and square

Add on both sides. -6 / 2 = -3 (-3)² = 9x² - 6x + 9 = 9/2 + 9

Step 5. Simplify both sides

Factor one side (the left in the example). The decomposed form will be (x - b / 2)². Add the terms that are similar to each other (on the right in the example). (X - 3) (x - 3) = 9/2 + 18/2 (x - 3)² = 27/2

Step 6. Find the square root of both sides

Don't forget to add the plus or minus sign (±) to the side of the constant x - 3 = ± √ (27/2)

Step 7. Simplify the root and solve for x

x - 3 = ± 3√ (6) ------- 2x = 3 ± 3√ (6) ------- 2

6 Ways to Factor Second Degree Polynomials (Quadratic Equations)

A polynomial contains a variable (x) raised to a power, called "degree", and several terms and / or constants. Decomposing a polynomial means reducing the expression to smaller ones that are multiplied together. It is a skill that is learned in algebra courses and can be difficult to understand if you are not at this level.

4 Ways to Solve Differential Equations

In a course on differential equations, the derivatives studied in an analysis course are used. The derivative is the measure of how much a quantity changes as a second varies; for example, how much the speed of an object changes with respect to time (in comparison to the slope).

3 Ways to Solve Algebraic Equations

First degree algebraic equations are relatively simple and quick to solve: most of the time two steps are enough to arrive at the final result. The procedure consists in isolating the unknown to the right or left of the equality sign using the addition, subtraction, multiplication or division operations.

3 Ways to Solve Linear Algebraic Equations with Multiple Unknowns

Linear equations with multiple unknowns are equations with two or more variables (usually represented by 'x' and 'y'). There are various ways to solve these equations, including elimination and substitution. Steps Method 1 of 3: Understanding the Components of Linear Equations Step 1.

5 Ways to Solve Equations that Have Variables on Both Sides

Solving equations with variables on both sides may seem daunting at first, but once you learn how to isolate the variable by moving it to one side of the equation, the problem will become much easier to handle. Here are some examples for you to review to practice this technique.