Here are new definitions to master: 

Let's consider the following ways of solving the quadratic equations:

• Vieta’s formulas
• Reversing Vieta's formulas
• Change of variables
• Factoring
• Completing the square
  
  Vieta’s formulas
  

The famous French scientist François Viet gave the world symbolic algebra. In the 16th century, he established relationships between the coefficients and the roots of a quadratic equation.

According to Vieta's formulas, the sum of the roots of the reduced quadratic equation of the form x² + px + q = 0 is equal to its second coefficient, taken with the opposite sign (-p), and the product is equal to its free term (q).

Solve a quadratic equation using Vieta's formula:

Determine the sum of the roots in the equations: (there are answers in the lower row)

Reversing Vieta's formulas

If numbers x₁ and x₂ fits the forms

x₁ + x₂ = -p,

x₁ * x₂ = q,

then they are the roots of the quadratic equation

x² + px + q = 0.

The way of solving a quadratic equation:

Change of variables

The way of solving the equation by changing the variable means reducing one equation to another. Let’s consider this algorithm more detailed:

Example:

2x4 +15x2 – 8 = 0

1. Let’s convert the biquadratic equations into quadratic one:
   2x4 + 15x2 – 8 = 2*(x2)2 + 15x2 – 8 = 0
   
2. We can add a new variable t = x2, then the equation will take the form:
   2t2 + 15t - 8 = 0
   
3. Now let's solve the obtained quadratic equation:
   

4. Substitute the obtained values of t into the equality t = x2 and get the system: 

5. The first equation has no roots. The roots of the second equation are:
 

Answer:  

Example:

2(x2 - x)2 – 5(x2 - x) – 3 = 0

Solution:

It is more convenient to solve this equation using the method of change of variables.

Let’s define that x2 – x = t,, then the equation will take the form: 2t² -5t - 3 = 0.

1. Solve the quadratic equation 2t² -5t - 3 = 0. 

2. Substitute the obtained values of t into the equality: 

3. Let’s solve the equation of the first system:
The discriminant is positive: D = 1+ 12 = 13> 0.
Then there are two roots of the equation:

4. Now let’s solve the equation of the second system:
The discriminant is negative (there are no roots): D = 1 – 2 = -1 < 0

Answer: 

Factoring

Sometimes it is necessary to factor a square trinomial ax² + bx + c = 0, using the formula: a * (x - x₁) (x - x₂).

The algorithm of factoring:

 

Factor the following equations:

Completing the square

We complete the square to transform a quadratic equation into a form that matches the abridged multiplication formulas:

(a + b)2 = a2 + 2ab + b2 ,

(a - b)2 = a2 - 2ab + b2
 

Prove Vieta's formulas

The sum of the roots of the reduced quadratic equation of the form x² + px + q = 0 is equal to its second coefficient, taken with the opposite sign (-p), and the product is equal to its free term (q).

Proof:

If the equation x² + px + q = 0 has roots x1 и x2.

Let’s prove, that x1 + x2 = -p, x1 * x2 = q.

1. According to the formulas of the roots of the quadratic equation: 

2. Let's find the sum of the roots: 

3. Find the product of the roots: 

This is what we set out to prove.