Question

X^2-13x+42=(x+a)(x+b)

Answer 1

D=169-168=1

x1= (13+1)/2= 7

x2= (13-1)/2= 6

a=-7,    b=-6

x²-13x+42=(x-7)(x-6)

Answer 2

To find the values of 'a' and 'b' that make the expression on the right side equal to the expression on the left side, we can start by expanding the right side and then comparing the coefficients with the left side.

The given expression on the left side is: x^2 - 13x + 42

And the expression on the right side is: (x + a)(x + b)

Expanding the right side:

(x + a)(x + b) = x^2 + ax + bx + ab

Now, we need to find 'a' and 'b' such that the coefficients of the expanded right side match those of the left side.

From the left side, we have: Coefficient of x^2: 1 Coefficient of x: -13 Constant term: 42

From the expanded right side, we have: Coefficient of x^2: 1 (This should match the left side) Coefficient of x: a + b (This should match the left side) Constant term: ab (This should match the left side)

Now we can equate the coefficients:

1. Coefficients of x^2: On the left side, it's 1. On the right side, it's also 1.

1 = 1

2. Coefficients of x: On the left side, it's -13. On the right side, it's a + b.

a + b = -13

3. Constant term: On the left side, it's 42. On the right side, it's ab.

ab = 42

Now, we have two equations:

1. a + b = -13
2. ab = 42

To find the values of 'a' and 'b', we can solve this system of equations. Let's proceed:

From equation 1, we can express 'a' as: a = -13 - b

Substitute this value of 'a' into equation 2:

(-13 - b) * b = 42 -13b - b^2 = 42 b^2 + 13b + 42 = 0

Now, we have a quadratic equation in terms of 'b'. We can solve for 'b' using factorization or the quadratic formula:

b^2 + 13b + 42 = 0 (b + 6)(b + 7) = 0

Now, either (b + 6) = 0 or (b + 7) = 0:

1. If b + 6 = 0, then b = -6.
2. If b + 7 = 0, then b = -7.

Now that we have the values of 'b', we can find the corresponding values of 'a' using the equation a = -13 - b:

1. If b = -6, then a = -13 - (-6) = -7.
2. If b = -7, then a = -13 - (-7) = -6.

So the two possible values for 'a' and 'b' are:

1. a = -7 and b = -6
2. a = -6 and b = -7

Both sets of values will satisfy the original equation. Therefore, the factorization of x^2 - 13x + 42 is:

(x - 7)(x - 6) or (x - 6)(x - 7)