Write the quadratic equation whose roots are -4 and 4 and whose leading coeffiecient is 3. Use the letter X to represent the variavble.

3x^2 - 48 = 0

Step-by-step explanation:

The roots of a quadratic equation whose roots are -4 and 4, means that valid solutions for the quadratic equation are -4 and 4.

The two forms we know for the quadratic equation are the factored binomials:

(ax + cy)(bx + dz) = 0

and the full equation

ax^2 + bx + c = 0

We will start with the factored form:

(x + 4) ( x - 4) = 0

Note that if we solve this, we get the roots:  x + 4 = 0  ==> -4 && x - 4 = 0 ==> 4

Now we apply foil to get the unfactored form:

(x + 4) (x - 4) = 0

x * x + x * -4 + 4 * x + 4 * -4 = 0

x^2 + -4x + 4x + -16 = 0

x^2 - 16 = 0

And now we want the leading coefficient to be 3.

3x^2 - 16  = 0

This looks like what we want; however, now with the leading 3, our solutions are no long +4 and -4.  So, we need to re-evaluate.

Going back to the binomials with the idea of a leading 3, let's rewrite them to still give us the solutions 4 and -4, but with the 3 included:

(3x + 12) (x - 4) = 0

Note, we simply multipled one binomial by 3 to maintain the roots of the original, but also incorporate the leading 3.

3x + 12 = 0              x - 4 = 0

3x = -12                        x = 4

x = -4

So now, we simply apply foil to this new equation of binomials to get our final quadratic equation:

(3x + 12) (x - 4) = 0

3x * x + 3x * -4 + 12 * x + 12 * -4 = 0

3x^2 + -12x + 12x + -48 = 0

3x^2 - 48 = 0

This is our quadratic equation we wish to use.

Note, we could have gotten this same equation by simply multiplying the original (x^2 - 16) = 0 by 3.

Cheers.

answer: (x+2) (x+3) are the binomials for your question : )