Quadratic Equations: In elementary algebra, a quadratic equation is any equation having the form where x represents an unknown, and a, b, and c are constants with a not equal to 0. If a = 0, then the equation is linear, not quadratic.
General Equation: x = -b ± √(b2 -4ac) / 2a OR
ax2 + bx + c = 0
Example: 2x2 + 3x + (-2) = 0
Here, a = 2, b = 3 and c = -2
Roots of the quadratic Equations:
Roots: Roots (or the values of ‘x’) are values which satisfies the given quadratic equation, i.e. when the values are put in the equation then left hand side(L.H.S) becomes equal to the right hand side (R.H.S) in the equation.
For quadratic equation of the general form: ax2 + bx + c = 0
Roots are given by α and β by the form (x- α)(x- β) = 0
How to Find Roots:
By the formula: x = -b ± √(b2 -4ac) / 2a
Just put the value of a, b, c by first comparing the equation with the standard equation (ax2 + bx + c = 0). After putting the values, two values of the x are obtained which are the requisite solutions.
Sum of Roots of a quadratic equation:
Given by: Sum of Roots = -b/a
Product of the roots of a quadratic equation:
Given by: Product of the roots: = c/a
Important Concept about the roots:
1. While finding roots if following conditions are satisfied, then the nature of the roots is as follows:
· When b2 -4ac <0, the roots are complex and unreal
· When b2 -4ac = 0, the roots are equal and real
· When b2 -4ac > 0, the roots are real and unequal
2. How to make a quadratic equation, when the roots are given (α,β):
Equation is formed as follows:
(x- α)(x- β) = 0
3. Maximum and Minimum values of a quadratic Equation:
· Quadratic equation will have minimum value whenever a>0
· Quadratic equation will have maximum value whenever a<0
4. Degree of an equation: It is the highest power of the variable. In case of quadratic equations it is always equal to 2.
5. Number of roots: Quadratic Equations always have 2 roots.
General Equation: x = -b ± √(b2 -4ac) / 2a OR
ax2 + bx + c = 0
Example: 2x2 + 3x + (-2) = 0
Here, a = 2, b = 3 and c = -2
Roots of the quadratic Equations:
Roots: Roots (or the values of ‘x’) are values which satisfies the given quadratic equation, i.e. when the values are put in the equation then left hand side(L.H.S) becomes equal to the right hand side (R.H.S) in the equation.
For quadratic equation of the general form: ax2 + bx + c = 0
Roots are given by α and β by the form (x- α)(x- β) = 0
How to Find Roots:
By the formula: x = -b ± √(b2 -4ac) / 2a
Just put the value of a, b, c by first comparing the equation with the standard equation (ax2 + bx + c = 0). After putting the values, two values of the x are obtained which are the requisite solutions.
Sum of Roots of a quadratic equation:
Given by: Sum of Roots = -b/a
Product of the roots of a quadratic equation:
Given by: Product of the roots: = c/a
Important Concept about the roots:
1. While finding roots if following conditions are satisfied, then the nature of the roots is as follows:
· When b2 -4ac <0, the roots are complex and unreal
· When b2 -4ac = 0, the roots are equal and real
· When b2 -4ac > 0, the roots are real and unequal
2. How to make a quadratic equation, when the roots are given (α,β):
Equation is formed as follows:
(x- α)(x- β) = 0
3. Maximum and Minimum values of a quadratic Equation:
· Quadratic equation will have minimum value whenever a>0
· Quadratic equation will have maximum value whenever a<0
4. Degree of an equation: It is the highest power of the variable. In case of quadratic equations it is always equal to 2.
5. Number of roots: Quadratic Equations always have 2 roots.
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