Question

1. Describe how factoring a quadratic expression ax2 + bx + c, where a ≠ 1, is different from factoring x2 + bx + c.

Answer 1

1.  Factoring a quadratic expression ax2 + bx + c, where a ≠ 1, is different from factoring x2 + bx + c because for the former type of expression you have to factor out the value of "a". Then, proceed to the factoring steps as usual.

2. To confirm the equations to be equal with the parent function we do as follows:
 (2x – 4)(x + 5) = 2x^2 + 10x - 4x - 20 = 2x^2 + 6x -20
(x – 2)(2x + 10) = 2x^2 +10x - 4x -20 = 2x^2 +6x - 20

3. The roots of the quadratic expression represents the values of x that would satisfy the expression. The x-intercepts are the values of x when y is equal to zero, it is where the plot touches intersects the x-axis.

Answer 2

Answer:

1. Dividing the expressions $ax^2+bx+c$ by a is a different step.

2. Yes, both of these factorization are correct. They are not complete because they can be factored further.

3. The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.

Step-by-step explanation:

1.

To factorize the quadratic expressions $ax^2+bx+c$ first we divide it by a. Then we factorize it same as $x^2+bx+c$.

It means all the steps of factoring a quadratic expressions $ax^2+bx+c$ and $x^2+bx+c$ are same except the first step, i.e., divide the expressions $ax^2+bx+c$ by a.

2.

The given quadratic expression is

$P(x)=2x^2+6x-20$

$(2x-4)(x+5)=2x(x+5)-4(x+5)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)$

$(x-2)(2x+10)=x(2x+10)-2(2x+10)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)$

The product of factors is equal to the given expression. It means both of these factorization are correct.

They are not complete because they can be factored further.

$(2x-4)(x+5)=2(x-2)(x+5)$

$(x-2)(2x+10)=(x-2)2(x+5)$

3.

If the factored form of a quadratic expression is defined as

$(x-a)(x-b)$

Then the related quadratic equation is

$(x-a)(x-b)=0$

$x=a,b$

The roots of the quadratic equation are a and b.

The related function is

$f(x)=(x-a)(x-b)$

The x-intercepts of the function are a and b because at x=a and x=b the value of function is 0.

The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.