Question

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

2. Two students factored 2x2 + 6x – 20. Keiko said that the factorization was (2x – 4)(x + 5). Ray gave the factorization as (x – 2)(2x + 10). Confirm that both of these factorizations are correct. Then explain why they are not complete.

3. Explain the relationship between the factors of a quadratic expression, the roots of the related quadratic equation, and the x-intercepts of the graph of the related function.

Answer 1

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.

Answer 2

It should be noted that in order to factorize the quadratic expression, one will have to divide it by a.

Factorization

The factorization of the quadratic expression ax² + bx + c is different from factoring x² + bx + c as one has to first divide it by a.

Secondly, the factorization by the students isn't complete because they can be factored further.

Lastly, the relationship between the factors of a quadratic expression, the roots of the related quadratic equation is that the roots are the x-intercept of the related function.

Learn more about factorization on:

https://brainly.com/question/22048677