Please Help?

Question

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.

Answer 1

(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.

Add 1/9 to each side.

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.

Answer 2

 

 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

  

Answer 3

(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).

Answer 4

(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