SAT Math Best shortcut on Quadratic Equations

#Shorts

This is a freebee if you get a question like this. Sometimes you get lucky. I didn't know about this tips until recently. -b/a to find the sum of the solutions of quadratic equations and c/a to find the product.

[konradfreymiller]

This is less of a “trick” and more just how quadratics and factoring works. Phrasing it as a trick undermines real understanding and concept mastery. You can get the answer by just recognizing the pattern of a quadratic in standard form.

When you are solving a quadratic, you want two numbers that add up to the “b” term and also multiply to the “c” term. For example, x^2+5x+6 turns into (x+2)(x+3), because 2+3=5 and 2*3=6. Consequently, the roots are -2 and -3 respectively.

Where the formula you just displayed and didn’t explain comes from the fact that if you have roots at -d and -e, (x+d)(x+e) expanded gives you x^2+ (d+e)x + de, where we see by structure that the “b” term is the addition of the the d and e term and the “c” term is the multiplication of the de term. In this case, the ROOTS are negative (-d and -e), so the addition of those is what the SAT question is looking for. Hence, the “b” term coefficient switches signs. The “c” term is the same because a negative times a negative is a positive.

The divide by “a” term just comes from factoring a out of the quadratic and now finding two numbers that add up to b/a and multiply to c/a.

No trick is necessary if you understand WHY we factor and how it relates to roots.

[createyourownfuture]

We can prove this very easily. Just take the quadratic formula and split it into the positive and negative versions. Those are the two solutions. Now add them to get the sum of the roots, and when you simplify you get -b/a. If you multiply them, you get the product of the roots, and it simplifies to c/a. This is how I first thought of proving it, but the pinned proof is also very smart.

[getthat]

Man you're a life saver, please keep it up!

[mokshsrivastava]

If you consider the two zeroes as x and y then just like

x + y = -b/a
xy = c/a
x-y = √Discriminant/a

[mryip]

FYI

α+β = -b/a is in the grade 10 syllabus in HK now.

[EKmaster]

So this gives the x coordinate of the vertex right?