Hello,

There are some questions that I am not sure how to do:

1) Solve by completing the square: 5 - 2x - 3x^2 = 0

2) Solve the equation: 5x^2 - 3 = 5x

3) Sketch the curves with the equations:

3a) y = 6x^2 - 7x - 3

3b) y = -x^2 + 4x + 5

Note: I know that in question 3 the equations need to be factorised, however I'm not sure how to do this. So please could you explain this to me in detail.

4a) Solve the equation 4x^2 + 12x = 0

4b) f(x) = 4x^2 + 12x + c, where c is a constant. Given that f(x) has equal roots, find the value of c and hence solve f(x) = 0.

Thank you very much in advance.

(1).

Completing the square Method :

The equation is 5 - 2x -3x2 = 0 ⇒ - 3x2 - 2x + 5 = 0.

Separate variables and constants aside.

Subtracting 5 from each side.

- 3x2 - 2x + 5 - 5 = 0 - 5

- 3x2 - 2x = - 5

Divide each side by negative 3.

-3x2/(-3) - 2x/(-3) = - 5/(-3)

x2 + 2x/3 = 5/3

To change the expression (x2 + 2x/3) into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

Here x coefficient = 2/3. So, (half the x coefficient)2 = (1/3)2= 1/9.

x2 + 2x/3 + 1/9 = 5/3 + 1/9

x2 + 2x/3 + (1/3)2 = (5*3 + 1)/9

(x + 1/3)2 = 16/9

x + 1/3 = ± √(16/9)

x + 1/3 = 4/3 and x + 1/3 = - 4/3

x = 1 and x = - 5/3

The value of x = 1 and x = - 5/3.

2).5x² - 3 = 5x

Rewrite the equation as 5x² - 5x -3=0

This is equation can be solved by using   .

Compare the given equation with general equation ax²+bx+c=0

4a) Given equation is 4x²  + 12x = 0

4x(x+3)=0

4x = 0 , x+3=0

The solution is x=0 , x= -3.

4b). Given equation is f(x)=4x²+12x+c

Given roots are equal than it seems satisy the condition b²-4ac=0

now compare given equation with general equation a=4 , b=12, c=c

b²-4ac = 12² - 4*4*c=0

144 -16c=0

c = 144/16

c = 9

Now to solve the eqution 4x² + 12x + 9 = 0

4x² + 6x + 6x + 9 = 0

2x(2x+3)+ 3(2x+3)=0

(2x+3)(2x+3)=0

(2x+3)²=0

2x+3=0

x= - 3/2 , - 3/2

answered Aug 26, 2014 by anonymous
edited Aug 26, 2014 by bradely

(3b).Graph y = -x2 + 4x + 5 making a table of values.

To find the y-intercept, substitute the value of x = 0 in the original equation.

y = -(0)2 + 4(0) + 5 = 5.

The y-intercept is 5, So the curve cross the y-axis at (0, 5).

To find the x-intercept, substitute the value of y = 0 in the original equation.

(0) = -x2 + 4x + 5

x2 - 4x - 5 = 0

x2 - 5x + x - 5 = 0

x(x - 5) + 1(x - 5) = 0

(x + 1)(x - 5) = 0

x = - 1 and x = 5.

The x-intercepts are -1 and 5, So the curve cross the x-axis at (-1, 0) and (5, 0).

Choose integer values for x and evaluate the function for each value. Graph the resulting coordinate pairs and connect the points with a smooth curve.

 x y = - x2 + 4x + 5 (x, y) -3 y = -(-3)2 + 4(-3) + 5 = - 9 - 12 + 5 = - 16 (-3, - 16) -2 y = -(-2)2 + 4(-2) + 5 = - 4 - 8 + 5 = - 7 (-2, - 7) -1 y = -(-1)2 + 4(-1) + 5 = - 1 - 4 + 5 = 0 (-1, 0) 0 y = -(0)2 + 4(0) + 5 = - 0 + 0 + 5 = 5 (0, 5) 1 y = -(1)2 + 4(1) + 5 = - 1 + 4 + 5 = 8 (1, 8) 2 y = -(2)2 + 4(2) + 5 = - 4 + 8 + 5 = 9 (2, 9) 3 y = -(3)2 + 4(3) + 5 = - 9 + 12 + 5 = 8 (3, 8) 4 y = -(4)2 + 4(4) + 5 = - 16 + 16 + 5 = 5 (4, 5) 5 y = -(5)2 + 4(5) + 5 = - 25 + 20 + 5 = 0 (5, 0) 6 y = -(6)2 + 4(6) + 5 = - 36 + 24 + 5 = - 7 (6, -7) 7 y = -(7)2 + 4(7) + 5 = - 49 + 28 + 5 = - 16 (7, -16)

Graph :

From the graph, the curve intersect the x-axis at (-1, 0) and (5, 0) and intersect the y-axis at (0, 5).

(3a).

Graph y = 6x2 - 7x - 3 making a table of values.

Choose integer values for x and evaluate the function for each value. Graph the resulting coordinate pairs and connect the points with a smooth curve.

 x y = 6x2 - 7x - 3 (x, y) -1 .5 y = 6(-1.5)2 – 7(-1.5) - 3 = 13.5 + 10.5 - 3 = 21 (-1.5, 21) -1 y = 6(-1)2 – 7(-1) - 3 = 6 + 7 - 3 = 10 (-1, 10) -0.5 y = 6(-0.5)2 – 7(-0.5) - 3 = 1.5 + 3.5 - 3 = 2 (-0.5, 2) 0 y = 6(0)2 – 7(0) - 3 = 0 - 0 - 3 = - 3 (0, -3) 0.5 y = 6(0.5)2 – 7(0.5) - 3 = 1.5 - 3.5 - 3 = - 5 (0.5, - 5) 1 y = 6(1)2 – 7(1) - 3 = 6 - 7 - 3 = - 4 (1, - 4) 1 .5 y = 6(1.5)2 – 7(1.5) - 3 = 13.5 - 10.5 - 3 = 0 (1.5, 0) 2 y = 6(2)2 – 7(2) - 3 = 24 - 14 - 3 = 7 (2, 7) 2.5 y = 6(2.5)2 – 7(2.5) - 3 = 37.5 – 17.5 - 3 = 17 (2.5, 17)

Graph :

From the graph, the curve intersect the x-axis at (-0.3, 0) and (1.5, 0) and intersect the y-axis at (0, -3).

CHECK :  Solve by factoring of 6x2 - 7x - 3 = 0.

6x2 - 7x - 3 = 0.

6x2 - 9x + 2x - 3 = 0.

3x(2x - 3) + (2x - 3) = 0.

(3x + 1)(2x - 3) = 0.

x = -1/3 = -0.3 and x = 3/2 = 1.5