Introduction to Quadratic Equations

Any equation of the form ax2 + bx + c = 0 is a quadratic equation, where a, b, and c are real numbers and a is not equal to 0 is called a quadratic equation.
If a and b are the roots for any equation, then (x – a) and (x – b) will be the factors of the equation.
An algebraic expression of the form ax2 + bx + c is called a quadratic function of x.
The graph of a quadratic equation is called a parabola.
The graph of the function f(x) = ax² + bx + c is concave upwards, when a > 0 and concave downwards, when a < 0.
If the graph cuts the x-axis, the roots of the equation will be real and unequal. Their values will be given by the abscissae of the points of intersection of the graph and the x-axis.
If the graph is tangent to the x-axis, the roots are real and equal.
If the graph has no points in common with the x-axis, the roots of the equation are imaginary and cannot be determined from the graph.
The sign of the expression ax² + bx + c is always the same as that of a, except when x lies between or at the roots of ax² + bx + c = 0.

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  • Q1

    The sum of two numbers is 15. If the sum of their reciprocal is 3/10. The two numbers are

    Marks:1
    Answer:

    10, 5.

    Explanation:

    Let two numbers are x and (15-x).Then,

     So numbers are 5 and 10 or 10 and 5.

     

     

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  • Q2

    The least value of ax2 + bx + c (a > 0) is

    Marks:1
    Answer:

    (4ac - b2)/4a.

    Explanation:

    If a > 0 then, ax2 + bx + c has a minimum value at x = -b/2a and is equal to (4ac - b2)/4a.

    If a < 0, then ax2 + bx + c has a maximum value at x = -b/2a and is equal to (4ac - b2)/4a.

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  • Q3

    The roots of the equation ax2 + bx + c = 0 will be reciprocal if

    Marks:1
    Answer:

    c = a.

    Explanation:
    Since roots are reciprocal, product of the roots = 1  c/a = 1  c = a. 

     

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  • Q4

    For a quadratic equation, is one root. The other root must be

    Marks:1
    Answer:

    Explanation:
     
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  • Q5

    If p and q are the roots of ax2 – 5x + c = 0 such that p + q = pq = 10, then the respective values of a and c are

    Marks:1
    Answer:

    (1/2, 5).

    Explanation:

    Sum of roots (p+q)= -(coefficient of x)/coefficient of x2 
                        p + q = (5/a)

                       10 = (5/a)

                           a = (1/2)

    Product of roots(pq) =(coefficient of x)/coefficient of x2 
                                 pq=c/a
                                 10=c/(1/2)
                          10 = 2c

                              c = 5

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