Answer:
[tex]-(x-4)(x+3)[/tex]
Step-by-step explanation:
[tex]12+x-x^2\\\\=(-12)(-1)+(-x)(-1)+(x^2)(-1)\\\\=-1(-12-x+x^2)\\\\=-(-12-x+x^2)\\\\=-(x^2-x-12)[/tex]
Here, the constant term 12 has a negative sign and the coefficient of x is -1. Hence, you are required to find two numbers whose product is 12 and difference is 1.
They are 4 and 3. Because 4-3 = 1 and 4x3 = 12
Now we rewrite the coefficient of x, -1 as -(4-3).
[tex]=-\{x^2-(4-3)x-12\}\\\\=-\{x^2-4x+3x-12\}\\\\=-\{x(x-4)+3(x-4)\}\\\\=-\{(x-4)(x+3)\}\\\\=-(x-4)(x+3)[/tex]
Solved Example:
If the constant term had a positive sign, you would have to find two numbers such that their product is the constant term, and the sum is equal to the coefficient of x.
Let's take an example:
[tex]x^2-5x+6[/tex]
Here, the sign of constant term 6 is positive and the coefficient of x is -5. So, we are required to find the two numbers whose product equals 6 and sum equals 5.
They are 3 and 2 because 3+2 = 5 and 3x2=6.
So now we rewrite -5, the coefficient of x as -(3+2).
[tex]x^2-(3+2)x+6\\=x^2-3x-2x+6\\=x(x-3)-2(x-3)\\=(x-3)(x-2)[/tex]
Answer:
a) See below
b) -(x + 3)(x - 4)
Step-by-step explanation:
To rewrite the expression 12 + x - x² in the form -(x² - x - 12), we need to factor out a negative sign from the expression.
Begin by placing the given expression in parentheses:
[tex](12 + x - x^2)[/tex]
Now, place the negative sign outside the parentheses and then divide each term inside the parentheses by -1.
When we factor out a negative sign from an expression, we are essentially multiplying the entire expression by -1. To cancel out this multiplication, we need to divide each term inside the parentheses by -1 to ensure that the resulting expression remains equivalent to the original one.
[tex]-\left(\dfrac{12}{-1}+\dfrac{x}{-1}-\dfrac{x^2}{-1}\right)[/tex]
This simplifies to:
[tex]-\left(-12-x-(-x^2)\right)\\\\\\-(12-x+x^2)[/tex]
Finally, rearrange the terms within the parentheses. This rearrangement does not change the expression's value, as we are only changing the order in which the terms appear (Commutative Property):
[tex]-(x^2 - x - 12)[/tex]
[tex]\dotfill[/tex]
To fully factorise the quadratic expression 12 + x - x², we can use the expression we derived in part a, which is -(x² - x - 12).
Factor the expression within the parentheses, x² - x - 12.
This is a quadratic expression in the form ax² + bx + c, where a = 1, b = -1 and c = -12.
Find two numbers that multiply to the product of a and c, and sum to up to b.
The product of a and c is 1 × -12 = -12. Two numbers that multiply to -12 and sum to -1 are -4 and 3. So, we can rewrite the quadratic expression inside the parentheses as:
[tex]-(x^2 - 4x + 3x - 12)[/tex]
Factor the first two terms and the last two terms within the parentheses separately:
[tex]-(x(x-4)+3(x-4))[/tex]
Factor out the common term (x - 4):
[tex]-(x+3)(x-4)[/tex]
So, the fully factorised form of the quadratic expression 12 + x - x² is:
[tex]\Large\boxed{\boxed{-(x+3)(x-4)}}[/tex]