The solution of a quadratic equation often comes down to finding of a discriminant. Whether depends on its value the equation will have roots and how many they will be. It is possible to bypass search of a discriminant only on a Vieta theorem formula if the quadratic equation is brought, that is has single coefficient at the senior multiplier.
Instruction
1. Define whether your equation is quadratic. It will be that if has an appearance: ах^2+bx+c=0. Here and, b and with are numerical constant multipliers, and x is a variable. If at the senior member (that is at what degree is higher therefore it х^2) there is a single coefficient, then it is possible not to look for a discriminant and to find equation roots on Vieta theorem which says that the decision will be the following: h1+ h2=-b; h1*kh2= with, where h1 and h2 - respectively equation roots. For example, the given quadratic equation: Õ^2+5Õ +6=0; On Vieta theorem system equations turns out: h1+ h2=-5; h1*kh2=6. Thus, it turns out h1=-2; h2=-3.
2. If the equation which is not given then not to avoid search of a discriminant. Determine it by a formula: D=b^2-4ас. If the discriminant is less than zero, the quadratic equation has no decisions if the discriminant is equal to zero, roots coincide, that is, the quadratic equation has only one decision. And only in case the discriminant is strictly positive, the equation has two roots.
3. For example, quadratic equation: 3х^2-18Õ +24=0. At the senior member there is a multiplier other than unit, therefore, it is necessary to find a discriminant: D = 18^2-4*3*24=36. The discriminant positive, therefore, the equation has two root.x1= (-b) +vD) / 2a= (18+6)/6=4; h2= (-b) - vD) / 2a= (18-6)/6=2.
4. Complicate a task, having taken such expression: 3х^2+9=12х-х^2. Transfer everything members to the left member of equation, without forgetting to replace the sign of coefficients, and in the right part leave zero: 3х^2+ х^2-12х +9=0;4х^2-12Õ +9=0. Now, looking at this expression, we can tell that it is square. Find a discriminant: D=(-12) ^2-4*4*9=144-144=0. The discriminant is equal to zero, so this quadratic equation has only one root which is determined by the simplified formula: h1.2=-v/2a=12/8=3/2=1.5.