Examples of using quadratic equations in real life

The length of the plot of land is 8 meters greater than the width. What is the size of this plot?

Solution:

Let the width of the plot be x, then the length is (x + 8), and the area is x * (x + 8). According to the problem condition, the area is 425 m². Let's make the equation for finding the area: x * (x + 8) = 425.

Let’s expand the brackets and we will get a quadratic equation x² + 8x = 425.

If the right part of the equation ≠ 0, it is necessary to exchange everything to the left side (we replace the right side digits to the left using the opposite signs). As a result, we get x² + 8x - 425 = 0. An equation of this type is called quadratic. Knowing how to solve such equations, we can find the x values, therefore, calculate the length and width of the plot of the land.

Having solved this equation, we get:

x = 17 (meters) - the width of the plot of the land, (x + 8) = 25 (meters) - its length.
Answer: 17m, 25m.
 

Definition:

If the equation fits the form ax² + bx + c = 0, where x is a variable, a, b, c are some numbers, and a ≠ 0, then it is a quadratic equation. Number a is the first coefficient, b is the second coefficient, and c is a free term.

Examples:

Let's consider examples of quadratic equations and define the constituent terms:
 

Solutions of quadratic equations

Before learning the many ways of solving quadratic equations, you need to explore such concepts:

a complete quadratic equation (all coefficients are not equal to zero) and an incomplete quadratic equation (some coefficients are equal to zero).
 

Answer if the equation is complete or incomplete?
 

Solving complete quadratic equations:

Here is the equation of the form ax² + bx + c = 0. Let’s make sure all the coefficients are not equal to zero: b ≠ 0 and c ≠ 0, a ≠ 0. Now we can start solving this complete quadratic equation. Firstly, we need to find the number of roots of the equation or their absence by calculating the discriminant (D). For an equation of the form ax² + bx + c = 0, where
a ≠ 0, b ≠ 0 or c ≠ 0, we can calculate the discriminant value by the formula: D = b² - 4ac (it is highly significant to remember!).

The algorithm of solving complete quadratic equations

Examples:

3x² + x + 2 = 0

Solution:

1. Determine the coefficients in the equation: a = 3, b = 1, c = 2.
2. Calculate the discriminant of the quadratic equation: D = b² - 4ac = (1)² - 4*3*2 = 1 - 24 = -23
3. Identify the discriminant value to deduce the properties of the roots: D = -23. It is less than zero. It means there won't be any roots of the equation.
   
   Answer: no roots.

x² - 6x + 9 = 0

Solution:

1. Find the coefficients in the equation: a = 1, b = -6, c = 9.
2. Determine the discriminant of the quadratic equation: D = b² - 4ac = (-6)² - 4*1*9 = 36 - 36 = 0
3. Use the minimum and maximum discriminant value to deduce the properties of the roots: D = 0 It is equal to zero. It means there will be one real root of the equation: x = (6) / 2*1 = 3.
   
   Answer: 3.
   
   The method of solving incomplete quadratic equations
   
   The equations that fit the form ax² + bx + c = 0, where b = 0 or c = 0, b = 0 and c = 0 are called incomplete quadratic equations. Let’s consider ways to solve the basic types of such equations: ax² + bx = 0, ax² + c = 0, ax² = 0.
    

Examples

-x² + x = 0

Solution:

1. Firstly, identify the coefficients of the equation: a = -1, b = 1, c = 0 - the equation is incomplete, as the free term is equal to zero.
2. Find the discriminant of the quadratic equation: D = b² - 4ac = (1) ² - 4 * (- 1) * 0 = 1 - 0 = 1
3. Use the discriminant value to deduce the properties of the roots: D = 1 - It is greater than zero. It means that the equation will have two roots.
4. Let’s determine the roots x₁ and x₂:
    

We can solve this incomplete quadratic equation using the form ax² + bx = 0:

-x² + x = 0

x * (-x + 1) = 0 → x = 0 → x = 0

   -x + 1 = 0 x = 1

Answer: 0, 1.


3x² - 27 = 0

Solution:

1. Let's find the coefficients of the equation: a = 3, b = 0, c = 27 - the equation is incomplete, as the coefficient b is equal to zero.
2. Calculate the discriminant of the quadratic equation: D = b² - 4ac = (0)² - 4*3*(-27) = 0 + 324 = 324
3. Identify the discriminant value to deduce the properties of the roots: D = 324 - It is greater than 0. It means the equation has two real roots.
4. Let’s calculate the roots x₁ and x₂:
   

Let’s solve the incomplete quadratic equation choosing the method based on the form ax² + c = 0:

3x² - 27 = 0

3 * (x² - 9) = 0

x² - 3² = 0 → (x - 3) = 0 → x = 3

   (x + 3) = 0 x = -3

Answer: -3, 3.


5x² = 0

Solution:

1. Let's determine the coefficients of the equation: a = 5, b = 0, c = 0 - the equation is incomplete, as the coefficient b and a free term equal zero.
2. Calculate the discriminant of the quadratic equation: D = b² - 4ac = (0)² - 4*5*(0) = 0 - 0 = 0
3. Analyze the discriminant value to deduce the properties of the roots: D = 0 - There will be one real root as the equation is incomplete and the discriminant equals 0.
   

We can apply the form ax² + c = 0 to solve this incomplete quadratic equation:

5x² = 0
x² = 0
x = 0

Answer: 0.