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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The sum of two numbers is 15. If the sum of their reciprocal is 3/10. The two numbers areMarks:1
Let two numbers are x and (15-x).Then,
So numbers are 5 and 10 or 10 and 5.
The least value of ax2 + bx + c (a > 0) isMarks:1
(4ac - b2)/4a.
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.
The roots of the equation ax2 + bx + c = 0 will be reciprocal ifMarks:1
c = a.
Explanation:Since roots are reciprocal, product of the roots = 1 c/a = 1 c = a.
For a quadratic equation, is one root. The other root must beMarks:1
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 areMarks:1
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
10 = 2c
c = 5