title: [Math Basics] Quadratic Equations
date: 2022-02-14 10:23:14
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- Mathematics
tags: - Mathematics
- Basics
- Quadratic Equations
Quadratic Equations#
A polynomial equation with only one unknown variable (one variable) and the highest power of the unknown variable term is 2 (quadratic) is called a quadratic equation. Quadratic equations can be rearranged into the general form ax² + bx + c = 0 (a≠0).
- Quadratic term: ax²
- Coefficient of the quadratic term: a
- Linear term: bx
- Coefficient of the linear term: b
- Constant term: c
Formulas (must be remembered)#
(a + b)(a - b) = a² - b²- Perfect square of a sum:
(a + b)(a + b) = a² + 2ab + b² - Perfect square of a difference:
(a - b)(a - b) = a² - 2ab + b² - Quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a - Memorize the squares of numbers from 1² to 20²
Difference of Squares Formula#
The structure of the perfect square formula:
- The left side is the product of two identical binomials, and the right side is a trinomial, which is the sum of the squares of the two terms in the left binomial, plus or minus twice their product.
- When the signs of the two terms on the left side are the same, all the terms on the right side are connected by the "+" sign. When the signs of the two terms on the left side are opposite, the square term on the right side is connected by the "+" sign, and then the product of the two terms is multiplied by 2 with a "-" sign (Note: The signs of the terms are not included here).
- The letters in the formula can represent specific numbers (positive or negative), as well as mathematical expressions such as monomials or polynomials.
(a + b)(a - b) = a² - b².
Find the values of a, b, and c in (x + 2)(x - 2) = 0.
x * (x - 2)+2 * (x - 2)- Obtain
x² - 2x + 2x - 4 - Simplify to
x² - 4 - Reverse the process of
(a + b)(a - b) = a² - b² - Result:
a = 1, b = 4, c = -4
Perfect Square of a Sum#
(a + b)(a + b) = a² + 2ab + b² is called the perfect square formula for the sum of two numbers.
Find the values of a, b, and c in (x + 2)(x + 2) = 0.
x * (x + 2)+2 * (x + 2)- Obtain
x² + 2x + 2x + 4 - Simplify to
x² + 4x + 4 - Reverse the process of
(a + b)(a + b) = a² + 2ab + b² - Result:
a = 1, b = 4, c = 4
Perfect Square of a Difference#
(a - b)(a - b) = a² - 2ab + b² is called the perfect square formula for the difference of two numbers.
Find the values of a, b, and c in (x - 2)(x - 2) = 0.
x * (x - 2)+-2 * (x - 2)- Obtain
x² - 2x - 2x + 4 - Simplify to
x² - 4x + 4 - Reverse the process of
(a - b)(a - b) = a² - 2ab + b² - Result:
a = 1, b = -4, c = 4
Solving Quadratic Equations#
Solve the quadratic equation: When is x equal to 0?
- Determine if there is a solution based on the discriminant
Δ = b² - 4ac - Δ > 0, there are two real solutions (usually one positive and one negative);
- Δ = 0, there is one solution;
- Δ < 0, there are no real solutions.
- Use the quadratic formula to solve the equation:
x = (-b ± √(b² - 4ac)) / 2a
Find the real solutions of the equation 2x² + 5x - 3 = 0.
Solution: Given a = 2, b = 5, c = -3.
Using the discriminant, we have Δ = 25 - 4 * 2 * (-3) = 25 - 8 * (-3) = 25 - (-24) = 25 + 24 = 49.
Using the quadratic formula, we have: x = (-5 + √49) / 2 * 2 = (-5 + 7) / 4 = 1 / 2 and x = (-5 - √49) / 2 * 2 = (-5 - 7) / 4 = -3
Advanced Method: Reverse Thinking#
Find the real solutions of the equation (x - 2)(1 - 3x) = 0.
Using the quadratic formula, we can calculate that the solutions are 1/3 and 2. However, by observing the equation (x - 2)(1 - 3x) = 0, we can see that the product of two numbers equals 0, which means either (x - 2) or (1 - 3x) equals 0.
We only need to calculate the corresponding values of x to obtain the solutions 1/3 and 2.
By using this method, we can also solve higher degree equations:
Find the real roots (solutions) of the equation x³ - 4x = 0.
Simplify the equation to: x(x² - 4) = 0, which gives us x = 0 and (x² - 4) = 0.
When x² = 4, we have x = ±2.
Therefore, the solutions are x₁ = 0, x₂ = 2, and x₃ = -2.
Square Number Table#
Pattern of Squares from 1 to 20#
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. If you look at this sequence of numbers, the difference between 1 and 4 is 3, the difference between 4 and 9 is 5, the difference between 9 and 16 is 7, and the difference between 16 and 25 is 9.
In other words, the differences between these numbers form a prime number sequence starting from 3.
Therefore, you can memorize the squares of 5, 10, and 15, and the differences between these numbers and the squares of the next digit. This way, you can quickly find the next square number!
For example, if you are asked for the square of 14, we know that the square of 15 is 225, and the difference between it and the next digit is 31, and the difference between it and the previous digit is 29. Therefore, we can deduce that the square of 14 is 225 - 29 = 196.
This makes it easy for us to remember!
Square Formulas Mnemonic#
Difference of Squares Formula#
The product of the sum and difference of two numbers is equal to the difference of their squares.
To transform the product into the sum and difference, the perfect square is not it.
Perfect Square Formula#
The square of the sum or difference of two numbers results in a trinomial with three terms.
The first square and the last square, with twice the product of the first and last terms in the middle.
The sum of the squares is connected, subtract first and then add the square of the difference.
Perfect Square Formula#
The first square and the last square, twice the first and last terms in the middle.
The sum of the squares is added again, subtract first and then add the square of the difference.
Relationship between Roots and Coefficients of Quadratic Equations#
Vieta's Formulas#
x₁ + x₂ = -(b / a)
x₁ * x₂ = c / a
Used to determine if the two real solutions of a quadratic equation are correct. The application of Vieta's formulas is based on the condition: b² - 4ac ≥ 0.
We can use the formulas of Vieta's formulas and known formulas to form a system of equations, which is a "system of linear equations with two variables and first degree".