Solving complete quadratic equations

To get formulas for calculating the roots of a complete quadratic equation, transform it:

The expression b² - 4ac is usually denoted by the letter D and is called the discriminant of the quadratic trinomial ax² + bx + с = 0.
Taking into account this notation, we will continue solving the quadratic equation


The last equation, and therefore the original one, may have two real roots, one root, or no real roots at all, depending on the sign of the discriminant D:
1. If D = b² - 4ac < 0 , then the quadratic equation ax² + bx + с = 0 has no real roots.
2. If D = b² - 4ac = 0, then the quadratic equation ax² + bx + с = 0 has a single real root x = :

3. If D = b² - 4ac > 0, then the quadratic equation ax² + bx + с = 0 has two real roots, which are calculated using the formulas

Let's show how to derive these formulas:


The last formula can be significantly simplified if b is divisible by 2, that is, b = 2k. Then the formula for the roots of the quadratic equation will look like this:

Here k = b/2. The resulting formula for the roots of a quadratic equation in the case of an even coefficient b can be rewritten without using the letter k:

or where D₁ = ()² - ac.
Obviously, the obtained formulas for the roots of complete quadratic equations can also be used to solve incomplete equations, although it is easier to use methods for solving incomplete quadratic equations.

Examples of solving quadratic equations

Example 1.
Solve the quadratic equation 4x² -28x + 49 = 0.
Solution.
Let's calculate the discriminant of the quadratic trinomial. We have a = 4, b = -28, c = 49.
Since b = -28 - is an even number, then we calculate the discriminant D₁ :
D₁ = ()² - ac = (-14)² - 4*49 = 196 - 196 = 0.
Therefore, the equation has a single root x = (b/2)/2 = 14/2 = 7. This equation can also be solved without calculating the discriminant by converting the quadratic trinomial using the reduced multiplication formula:
4x² -28x + 49 = 0 <=> (2x - 7)² = 0 <=> 2x = 7 <=> x = 7/2.
Answer: 7/2.

Example 2.
Solve the equation Solution.
Let's bring the left side of the equation to a common denominator:


Multiplying both parts of the equation by -6, we get x² + 3x = 0. We will solve this incomplete quadratic equation by factoring it:

Answer: -3, 0.

Example 3.
Solve the equation Solution.
Let's bring the left and right sides of the equation to a common denominator:


Multiplying both parts of the equation by 15, we get:
6x² + 3x = 20x-10 <=> 6x² + 3x - 20x + 10 = 0 <=> 6x² - 17x + 10 = 0.
Let's calculate the discriminant of the quadratic trinomial: a = 6, b = -17, c = 10.
D = b² - 4ac = (-17)² - 4*6*10 = 289 - 240 = 49 > 0, therefore, the equation has two real roots:

Answer: 5/6, 2.

Example 4.
Solve the equation
Solution.
Let's calculate the discriminant of the quadratic trinomial. We have a = 1, b = 2√2, c = 1.
Since b = 2√2, that is, b is divisible by 2 (b/2 = √2), let's calculate the discriminant D₁:
D₁ = ()² - ac = (√2)² - 1*1 = 1 > 0. Therefore, the equation has two real roots.

Answer: -√2-1, -√2+1.

Example 5.
Solve the quadratic equation
Solution.
Multiply the left and right sides of the equation by 6:


Let's calculate the discriminant of the resulting quadratic trinomial. We have a = 3, b = -6, c = 2.
Since b = -6, that is, b is divisible by 2 (b/2 = -3), let's calculate the discriminant D₁:
D₁ = (b/2)² - ac = 3² - 3*2 = 3 > 0. Therefore, the equation has two real roots.


Answer: