Find the roots of the following quadratic equations by factoring.
y=x^2+9x-36
I got (x+12)(x-3) but on the answer key it says that the answer is this, however it renames the x values with opposite values.
For example, the answer on the review for it has my answer, but the final answer is x=-12 and x=3. Is this because numbers in the parenthesis are always opposite of what they are? Like a number that is positive will be a negative, and vice-versa?
Thank you.
Nevermind got it! it's because it's supposed to equal 0 so you have to solve the equation to make it zero. Basically by doing the inverse of the number but not really.
First of all , let me point out a discrepancy in the wording of your question
The way your equation is written as y = x^2 + 9x - 36
it should have asked for the x-intercepts of the function
To do that , we realize that for an x-intercept, the value of y = 0
so your function becomes the quadratic equation
x^2 + 9x - 36 = 0 , which factors to
(x+12)(x-3) = 0 , .............. you had that.
Please agree with my following argument:
If two numbers are multiplied and the answer is zero, then one of the multipliers, or both, must have been zero.
But we don't know which one, so we say:
x+12 = 0 -----> x = -12
or
x-3 = 0 -----> x = 3
What you did was FACTOR the expression, but it asked for the roots.
Can you see why the signs would always change?
(Because we are taking the constant to the other side of the equal sign)
e.g. if one of factors had been 3x + 5
we would say:
3x+5 = 0
3x = -5
x = -5/3
To find the roots of a quadratic equation by factoring, you need to set the equation equal to zero and factor the equation to obtain two binomial expressions.
In the case of the quadratic equation y = x^2 + 9x - 36, you correctly factored it as (x + 12)(x - 3). This means that if you set each factor equal to zero, you can solve for the values of x that make the equation true:
x + 12 = 0 or x - 3 = 0
Solving these equations individually, you get:
For x + 12 = 0, subtracting 12 from both sides gives x = -12.
For x - 3 = 0, adding 3 to both sides gives x = 3.
So, according to your factoring, the roots of the quadratic equation are x = -12 and x = 3, which matches the given answer key.
Now, regarding the renaming of x values with opposite signs, it seems you are referring to the solutions being opposite of what appears in the factored form. The reason for this is the concept of opposite signs in factored quadratic equations.
When you expand the factored form, (x + 12)(x - 3), you will get x^2 + 9x - 36, which is the original quadratic equation. Notice that the term with x^2 remains positive, regardless of the signs of the factors.
However, when you set the factors equal to zero, you reverse the signs, as you are solving for the values that make each factor zero. In other words, you are finding the x-values where the graph of the quadratic equation intersects the x-axis, which happens when y (or in this case, the equation) equals zero.
To summarize, the roots of the quadratic equation y = x^2 + 9x - 36 are x = -12 and x = 3, which is consistent with your factoring and the given answer key. The renaming of x-values with opposite signs occurs when solving for the values that make the factors equal to zero.