# What is x if x 7 12

## Solve quadratic equations with the p / q formula

bettermarks »Math book» Algebra and functions »Functions and their representations» Solving methods for quadratic equations »Solve quadratic equations with the p / q formula

### Derivation of the pq formula

Solution formula for a quadratic equation in normal form pq formula:

The pq formula arises from that of a quadratic equation through. For :

### Solving quadratic equations

Solution formula for a quadratic equation in normal form pq formula:

You insert and into the pq formula: and

### Determine the number of solutions with the discriminant

Discriminant D for the pq formula:

If you consider the discriminant D der, you can state how many solutions one has. If D> 0, the equation has two solutions.

If D = 0, the equation has a solution.

If D <0, the equation has no solution.

### Theorem of Vieta

Francois Viète (lat. Vieta) discovered the connection between p and q and the solutions and the quadratic equation: You can use Vieta's theorem to check whether two values are solutions of a given quadratic equation. ()

Are and solutions to the equation? You put and in the equations: Therefore, at least one of the values is not a solution to the quadratic equation. You can use Vieta's theorem to “guess” the solutions of a quadratic equation.

What solutions does the equation have? You plug in the equations for p and for q and look for numbers for and that satisfy both equations: The two factors 2 and 3 of 6 are solutions to the quadratic equation. You confirm this by inserting:

### Derivation of the Vieta theorem

For a quadratic equation in normal form (), Vieta's theorem applies: and.

You can derive that from the. If and the solutions are that, then:

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