Complete the square x2 5x 4 Complete the square x2 8x 2

Complete the square: x^2 + 5x + 4. Complete the square: x^2 - 8x + 2. Complete the square: x^2 + 5x - 2. You are given the coordinates of the vertex (-4, -9) and of a point (2, -7) a parabola. Find the equation of the parabola. You are given the coordinates of the vertex (1, -4) and of a point (2, 6) on a parabola. Find the equation of the parabola.

Solution

1. x²+ 5x + 4 = x²+ 2*5/2x + (5/2)² + 4 - (5/2)² = ( x + 5/2)² + 4 - 25/4 = ( x + 5/2)² - 9/4.

2. x² - 8x + 2 = x² - 2*4x + (4)² + 2 - (4)² = ( x -4)² + 2 - 16 = (x -4)² - 14.

3. x²+ 5x - 2 = x²+ 2*5/2x + (5/2)² -2 - (5/2)² = ( x + 5/2)² -(2 + 25/4) = ( x + 5/2)² - 33/4.

4. The vertex form of the equation of a parabola with vertex at (h,k) is y = a (x - h)² + k , where a is an arbitrary real number. Here h = -4 and k = -9. Therefore, the equation of the parabola changes to y = a ( x +4)² - 9. Further, the parabola passes through the point ( 2, -7), so on substituting x = 2 and y = -7 in the equation of the parabola, we have -7 = a ( 2 + 4)² - 9 or, 36a - 9 = -7 or, 36a = -7 + 9 = 2 so that a = 2/36 = 1/18. Finally, the equation of the required parabola is y = 1/18( x + 4)² - 9.

5. The vertex form of the equation of a parabola with vertex at (h,k) is y = a (x - h)² + k , where a is an arbitrary real number. Here h = 1 and k = -4. Therefore, the equation of the parabola changes to y = a ( x -1)² - 4. Further, the parabola passes through the point ( 2, 6), so on substituting x = 2 and y = 6 in the equation of the parabola, we have 6 = a ( 2 - 1)² - 4 or, a - 4 = 6 or, a = 6 + 4 = 10 . Finally, the equation of the required parabola is y = 10( x -1)² - 4.