Question

9. Show the expression x² - x - 6 in factored

form and explain what the solution would mean
for the equation. Show your work.

Answer 1

Given:-

• A expression x² - x - 6 .

To find:-

• The factored form of the expression.
• What would solution mean for equation?

Answer:-

Given expression to us is ,

`\implies x^2 - x - 6 \\`

Here we can write " - x " as the sum of -3x and x .

So we can write the given expression as ,

`\implies x^2 - 3x + 2x - 6\\`

`\implies x ( x - 3) + 2(x -3) \\`

`\implies (x+2)( x-3)\\`

Hence the factored form of the expression is (x+2)(x-3) .

By zeroes we mean the values , which when put in the equation at the place of variable here x , the value of expression becomes zero .

For finding the zeroes, we need to equate the given expression with 0 as ,

`\implies (x+2)(x-3) = 0 \\`

Here we can two values of x as ,

`\implies \underline{\underline{ x = -2 , 3 }}\\`

This means when we put x = -2 or x = 3 in the given equation , the equation becomes 0 .

and we are done!

Answer 2

The given expression in factored form is (x + 2)(x - 3).

If the expression is set to zero, the solutions x = -2 and x = 3 are the roots of the graph of the equation.

Explanation:

To factor a quadratic in the form ax² + bx + c, begin by finding two numbers that multiply to ac and sum to b.

Given quadratic:

`x^2-x-6`

Therefore:

`\implies ac=1 \cdot -6=-6`

`\implies b=-1`

Two numbers that multiply to -6 and sum to -1 are -3 and 2.

Rewrite b as the sum of these two numbers:

`\implies x^2-3x+2x-6`

Factor the first two terms and the last two terms separately:

`\implies x(x-3)+2(x-3)`

Factor out the common term (x - 3):

`\implies (x+2)(x-3)`

If the given expression is a function, then:

`f(x)=(x+2)(x-3)`

To find the solutions of the function, set it to zero and apply the zero-product property:

`(x+2)=0 \implies x=-2`

`(x-3)=0 \implies x=3`

Therefore, the solutions of the equation are x = -2 and x = 3.

These are the roots (x-intercepts) of the graph of the function.