Quadratic Functions

Question 10: The quadratic equation x 2 −4x−1=2p(x−5) , where p is a constant, has two equal roots. Calculate the possible values of p. Solution: Question 11: Find the range of values of k for which the equation x 2 −2kx+ k 2 +5k−6=0 has no real roots. Solution: Question 12: Find the range of … Read more

Question 7: Find the value of p if one root of x 2 +px+8=0 is the square of the other. Solution: Question 8: If one root of 2 x 2 +px+9=0 is twice the other, find the values of p. Solution: Question 9: The roots of the equation 6 x 2 +hx+1=0 are α and β, whereas 3α and 3β are the roots … Read more

Question 4: If the roots of 2 x 2 +4x−1=0 are α and β, find the equations whose roots are (a)  α 2 , β 2 (b) α−β, β−α Solution: Question 5: Write and simplify the equation whose roots are double the roots of 3 x 2 −5x−1=0 , without solving the given equation. Solution: Question 6: Write and simplify … Read more

Question 1: Solve the following quadratic equations by factorisation. (a)  x 2 −5x−10=−4 (b) 3−x−2 x 2 =0 (c) 11a=2 a 2 +12 (d)  2x+7 3x−2 =x Solution: Question 2: Solve the following quadratic equations by completing the square. (a) 5 x 2 +10x−3=0 (b) 2 x 2 −5x−6=0 Solution: Question 3: Solve the following quadratic equations by using … Read more

Example 1Express y = 5 + 4 x − x 2   in the form y = a − ( x + b ) 2   , where a, b, and c  are constants. Hence state the maximum  value of y and the value of x at which it occurs. Sketch the curve y = … Read more

Graphing Quadratic Function If you are asked to sketch the graph of a quadratic function, you need to show a. the shape of the graph b. the maximum/minimum point of the graph c. the x-intercept of the graph d. the y-intercept of the graph Example Sketch the curve of the quadratic function f ( x … Read more

Example 1 Find the maximum or minimum value of each of the following quadratic function by completing the squares.  In each case, state the value of x at which the function is maximum or minimum.  And also, state the maximum or minimum point and axis of symmetry for each case. (a) f ( x ) = … Read more

Steps to convert general form of Quadratic Function into completing the square form General form of quadratic function : f ( x ) = a x 2 + b x + c Completing the square form : f ( x ) = a ( x + p ) 2 + q Step 1 : Make … Read more

Example 1State the maximum or minimum value of   for each of the following quadratic function and state the value of x at which the function is maximum or minimum.  Find the maximum or minimum point and finally state axis of symmetry for each case. (a) f ( x ) = − 2 ( x … Read more

Maximum and Minimum Point A quadratic functions f ( x ) = a x 2 + b x + c can be expressed in the form f ( x ) = a ( x + p ) 2 + q by the method of completing the square. The minimum/maximum point can be determined from the … Read more