## 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.

To Access the full content, Please Purchase

• Q1

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

Marks:1

10, 5.

##### Explanation:

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

So numbers are 5 and 10 or 10 and 5.

• Q2

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

Marks:1

(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.

• Q3

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

Marks:1

c = a.

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

• Q4

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

Marks:1

##### Explanation:

• 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

(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