A quadratic equation is a function that is written in the form of ax² + bx + c = 0 where a, b, c are real numbers. The solutions of a quadratic equation are known as zeros or roots of the equation. In the previous lessons, we have learned about what is a quadratic equation and how to solve quadratic equations. Now on this page, we will come to know the general properties of the quadratic equations.
Also, See:
How to Solve a Quadratic Equation?
Here we have provided the steps to solve the quadratic equations.
i. First write the equation in the standard form of ax² + bx + c = 0.
ii. Factorize the left side of the equation.
iii. Now convey each of the two factors to 0 and solve the equation.
iv. The results of the equation are called roots of the quadratic equation.
General Properties of Quadratic Equation | Quadratic Functions and their Properties
1. The zeros of the quadratic function that is written in the form of ax² + bx + c = 0. It has two values of x. The equation ax² + bx + c = 0 is called quadratic equation.
2. The general form of the quadratic equation is ax² + bx + c = 0.
3. Find the zeros of the equation or solving the quadratic equation that is the same. There are three methods to find the zeros of the quadratic equation. They are completing the square, quadratic formula, factoring.
4. The sum of the roots of quadratic equation in the general form ax² + bx + c = 0 is -b/a.
5. The product of the roots of the quadratic equation in the standard form ax² + bx + c = 0 is c/a.
6. If two zeros of a quadratic equation ax² + bx + c = 0 are reciprocal to each other then c = a.
7. If two zeros of a quadratic equation ax² + bx + c = 0 are equal in magnitude, but opposite in sign, then their sum is b = 0.
8. If two roots of Q.E are irrational then the two roots will appear in conjugate pairs.
9. The formula to find the roots of the quadratic equation is
x² – (sum of the roots)x + product of the roots = 0
10. If the two roots of the quadratic equation are imaginary then the graph will not intersect at the x-axis.
11. If the roots of the quadratic function are x-coordinates of the locus of the point of the parabola parts x-axis.
12. The graph of the quadratic equation is a parabola.
13 x-coordinate of the vertex of the parabola is -b/2a and the vertex is [-b/2a, f(-b/2a)]
14. If the discriminant b² – 4ac = 0 then the roots are real, equal, and rational.
15. If b² – 4ac > 0 then the roots are real, distinct, and rational.
16. If b² – 4ac < 0 then the roots are imaginary.
FAQs on Basic Properties of Quadratic Equations
1. What is the General form of a quadratic equation?
The general form of a quadratic function is f(x)=ax2+bx+c.
2. What are the three properties to be used when solving quadratic equations?
There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square.
3. What are the two properties of the quadratic equation?
The two properties of the quadratic equations are as follows,
1) The graph of a quadratic function is always a parabola that either opens upward or downward
2) The domain of a quadratic function is all real numbers