Solve each of the following equations using the method indicated:

(i) -3x^2 +10x+8=0 by factorization;

(ii) 2x^2 + 9x-5=0 by completing the square. (6 marks)DICT MOD 1 July 2020

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(i) To solve the equation -3x^2 +10x+8=0 by factorization, we start by setting the expression -3x^2 +10x+8 equal to zero and then factoring the quadratic. We can use the quadratic formula to find the roots of the equation, which are the values of x that make the equation true.

-3x^2 +10x+8=0

To factor this quadratic, we can use the factorization method, which involves expressing the quadratic as the product of two binomials. One way to do this is to rewrite the quadratic as follows:

-3x^2 +10x+8=0

(-3x^2 +10x) + 8=0

(-3x^2 +10x) = -8

We can now factor the left-hand side of the equation by finding two numbers whose product is -8 and whose sum is 10. The numbers that satisfy these conditions are -4 and 2, so we can rewrite the left-hand side of the equation as follows:

(-3x^2 +10x) = (-4)(2)

(-3x^2 +10x) = (-4x +2)(2)

We can now rewrite the original equation as follows:

(-4x +2)(2) + 8=0

(-4x +2)(2) = -8

This equation can be simplified to:

-4x +2 = -4

-4x = -2

x = 1/2

Therefore, the solution to the equation is x = 1/2.

(ii) To solve the equation 2x^2 + 9x-5=0 by completing the square, we start by setting the expression 2x^2 + 9x-5 equal to zero and then completing the square to find the roots of the equation.

2x^2 + 9x-5=0

To complete the square, we need to rewrite the quadratic as a perfect square trinomial of the form (x + b)^2 = x^2 + 2bx + b^2. We can do this by adding and subtracting the square of half of the coefficient of the x term. In this case, the coefficient of the x term is 9, so we add and subtract (9/2)^2 = (9/2)*(9/2) = 9/4 = 2.25. This gives us the following equation:

2x^2 + 9x-5=0

2x^2 + 9x – 2.25 + 2.25 -5=0

(2x^2 + 9x – 2.25) + (2.25 -5)=0

(2x^2 + 9x – 2.25) = -(2.25 -5)

(2x^2 + 9x – 2.25) = -(-0.25)

(2x^2 + 9x – 2.25) = 0.25

We can now rewrite the left-hand side of the equation as a perfect square trinomial by completing the square:

(2x^2 + 9x – 2.25) = (x + (9/2))^2

(x + (9/2))^2 = 0.25

x + (9/2) = sqrt(0.25)

x + (9