Instruction
1
How to find the discriminant? There is a formula of its location: D = b2 - 4ac. In this case, if D > 0, the equation has two real roots, which are calculated by the formulas:
x1 = (-b + VD)/2a,
x2 = (-b - VD)/2a,
where V means square root.
x1 = (-b + VD)/2a,
x2 = (-b - VD)/2a,
where V means square root.
2
To understand the formula in action, solve a few examples.
Example: x2 - 12x + 35 = 0, in this case a = 1, b - (-12), free member - + 35. Find the discriminant: D = (-12)^2 - 4*1*35 = 144 - 140 = 4. Now find the roots:
X1 = (-(-12) + 2)/2*1 = 7,
x2 = (-(-12) - 2)/2*1 = 5.
When a > 0, x1 < x2, for α < 0, x1 > x2, which means if the discriminant is greater than zero: there are real roots, the graph of a quadratic function crosses the axis OX in two places.
Example: x2 - 12x + 35 = 0, in this case a = 1, b - (-12), free member - + 35. Find the discriminant: D = (-12)^2 - 4*1*35 = 144 - 140 = 4. Now find the roots:
X1 = (-(-12) + 2)/2*1 = 7,
x2 = (-(-12) - 2)/2*1 = 5.
When a > 0, x1 < x2, for α < 0, x1 > x2, which means if the discriminant is greater than zero: there are real roots, the graph of a quadratic function crosses the axis OX in two places.
3
If D = 0, then one solution:
x = -b/2a.
If the second coefficient of the quadratic equation, b represents an even number, it is advisable to find the discriminant divided by 4. The formula will be as follows:
D/4 = b2/4 - ac.
For example, 4x^2 - 20x + 25 = 0, where a = 4, b = (- 20) = 25. Where D = b2 - 4ac = (20)^2 - 4*4*25 = 400-400 = 0. Square trinomial has two equal roots, find them according to the formula x = -b/2a = - (-20)/2*4 = 20/8 = 2,5. If the discriminant is zero, then there is one real root, the graph of the function crosses the axis OX in one place. Thus, if a > 0, the graph is above the x-axis, and if a < 0, and below this axis.
x = -b/2a.
If the second coefficient of the quadratic equation, b represents an even number, it is advisable to find the discriminant divided by 4. The formula will be as follows:
D/4 = b2/4 - ac.
For example, 4x^2 - 20x + 25 = 0, where a = 4, b = (- 20) = 25. Where D = b2 - 4ac = (20)^2 - 4*4*25 = 400-400 = 0. Square trinomial has two equal roots, find them according to the formula x = -b/2a = - (-20)/2*4 = 20/8 = 2,5. If the discriminant is zero, then there is one real root, the graph of the function crosses the axis OX in one place. Thus, if a > 0, the graph is above the x-axis, and if a < 0, and below this axis.
4
When D < 0 the real roots are there. If the discriminant is less than zero, then there is no real roots, only complex roots, the function graph does not cross the axis OX. Complex numbers - extension of the set of real numbers. A complex number can be represented as a formal sum of x + iy, where x and y are real numbers, i is imaginary unit.
Note
The equation of the form ax2+bx+c=0 a necessary condition is the inequality and zero.
If a is equal to one, the equation referred to above.
If a is not equal to one, that nepreodolim. If one of the coefficients b, C or both are equal , then the quadratic equation is called incomplete
If a is equal to one, the equation referred to above.
If a is not equal to one, that nepreodolim. If one of the coefficients b, C or both are equal , then the quadratic equation is called incomplete