c = product of roots, b = sum of roots, a = scaling

When factoring an exponential equation (ax2+bx+c into (x-d)(x-e)), the signs and magnitudesof b and c can provide a lot of information that is useful for factoring.

Example Equation: x² + 3x − 10

C

C should always be the starting place any time a student factors. It gives direct and specific information about the constant terms of the pair of binomials that would be created by factoring,

C is the equal to the product of the two constants. This means that:

• Magnitude: The magnitude of C is equal to the product of the magnitude of the two constants in the binomials
• Sign: If the sign of C is positive, then the constants in the binomial have the same sign (two negatives or two positives). If the sign of C is negative, then the constants in the binomial have opposite signs (one positive and one negative).

In the example equation above, the magnitude of C is 10, meaning that the constants would be factors of 10. The sign of C is negative, which means that the constants have opposite signs.

B

B is equal to the sum of the two constants. Combining this with what is known from C:

• If the signs of the constants are the same, then the sign of B is the same as the signs of both constants
• If the signs of the constants are opposites, then the sign of B is the same as the constant with the larger magnitude (the bigger number regardless of sign)

A

A is the product of the coefficients in front of the variable terms in the binomials. Most ACT quadratic equations will not have a coefficient in from of the variable term (the coefficient will just be one), so B and C are the main things students should be comfortable with translating.

However, if the A term is something other than 1, it’s worth knowing the following things:

• Normally A will be a perfect square: 1, 4, 9, 16. Meaning that the signs in front of the variable term will be the same in each binomial.
• If A is not a perfect square: It will usually be an odd, prime number such as 3, 5, or 7, as then the coefficients in front of the variables have to be that number and 1.
• A does not affect C, but it will scale the terms that add up to B: since C is just the product of the two variable terms, it is not affected by coefficients on the x term. However, it will affect the variable terms that sum to b.
  • If A is a perfect square, then the terms that sum to B will be scaled evenly
  • If A is not a perfect square, then one term will be scaled by a larger factor than the other.