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Question

Solve each quadratic equation by completing the square and express your answer(s) as EXACT VALUES.

a) x^2 + 6x - 4 = 0

b) 3x^2 - 24x = 17

 

- Explain in detail how you would check your answer to question a) by using a graph.

Answer 1

Completing square method

a) The equation x² + 6x - 4 = 0

Separate variables and constants aside by adding 4 to each side.

x² + 6x - 4 + 4= 0 + 4

x² + 6x = 4

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the equation.

 Here x coefficient = 6. so, (half the x coefficient)² = (6/2)²= 9

Add 9 to each side

x² + 6x + 9 = 4 + 9

(x + 3)²= 13

Take square root both sides.

x + 3 = ± √13

Solution x = - 3 + √13 and x = - 3 - √13.

Answer 2

b) The equation 3x² - 24x = 17

Here x² coefficient is 3 , for perfect square make x² coefficient 1 by dividing each side by 3.

(3x² - 24x)/3 = 17/3

x² - 8x = 17/3

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the equation.

 Here x coefficient = -8. so, (half the x coefficient)² = (-8/2)²= 16

Add 16 to each side

x² - 8x + 16 = 17/3 + 16

(x - 4)² = ( 17 + 48)/3

(x - 4)² = 65/3

Take square root both sides.

x - 4 = ± √(65/3)

Solution x = 4 + √(65/3) and x = 4 - √(65/3).

Answer 3

Check the answer by using graph

a) Solutions of x² + 6x - 4 = 0 are x = - 3 + √13 and x = - 3 - √13.

x = - 3 + 3.6 and x = - 3 - 3.6

x = 0.6 and x = - 6.6

Solutions are x intercepts of function.

The quadratic function represents a parabola.

The parabola equation y = x² + 6x - 4

Make the table of values to find ordered pairs that satisfy the equation.

Choose random values for x and find the corresponding values for y.

x	y = x2 + 6x - 4	(x, y)
- 6	y = (-6)2 + 6(-6) - 4 = -4	(- 6, -4)
-4	y = (-4)2 + 6(-4) - 4 = -12	(-4, -12)
-2	y = (-2)2 + 6(-2) - 4 = -12	(-2, -12)
0	y = (0)2 + 6(0) - 4 = -4	(0, -4)
1	y = (1)2 + 6(1) - 4 = 4	(1, 4)

Draw the coordinate plane.

Plot the points found in the table.

Connect the plotted points with smooth curve.

Observe the graph x intercepts are 0.6 and -6.6.