ACT Math: Sum and Product of Roots—Vieta's Formulas Shortcut

For a quadratic ax²+bx+c=0 with roots r and s: Sum of roots (r+s)=-b/a. Product of roots (rs)=c/a. These formulas are shortcuts that avoid finding roots explicitly. Instead of solving x²+5x+6=0 using the quadratic formula, notice sum=-5/1=-5 and product=6/1=6. What two numbers sum to -5 and multiply to 6? Answer: -2 and -3. So roots are -2 and -3. This method is faster than the quadratic formula and reveals the structure of a polynomial instantly.

Why it works: The quadratic (x-r)(x-s)=x²-(r+s)x+rs. Comparing to x²+bx+c, we see -(r+s)=b, so r+s=-b, and rs=c.

Use 1 (Find roots given coefficients): For x²-7x+12=0, sum=7, product=12. What two numbers sum to 7 and multiply to 12? Answer: 3 and 4. Roots are 3 and 4. Use 2 (Reconstruct polynomial given roots): If roots are 5 and -2, reconstruct the polynomial. Sum=5+(-2)=3, product=5×(-2)=-10. Polynomial is x²-3x-10=0 (or y=x²-3x-10). General form: x²-(sum)x+(product)=0. This second use is powerful: given two roots, you can instantly write the polynomial without expanding (x-r)(x-s).

For cubic ax³+bx²+cx+d=0 with roots r, s, t: r+s+t=-b/a, rs+rt+st=c/a, rst=-d/a. The pattern extends.

Problem 1: For x²+6x+8=0, find roots using Vieta. Sum=-6, product=8. Numbers that sum to -6 and multiply to 8: -2 and -4. Roots: -2, -4. Problem 2: For x²-10x+21=0, find roots. Sum=10, product=21. Numbers: 3 and 7. Roots: 3, 7. Problem 3: Reconstruct polynomial with roots 6 and 1. Sum=7, product=6. Polynomial: x²-7x+6=0. Problem 4: For 2x²+10x+12=0 (divide by 2: x²+5x+6=0), find roots. Sum=-5, product=6. Numbers: -2, -3. Roots: -2, -3. Problem 5: Given roots -1 and 4, write the polynomial. Sum=3, product=-4. Polynomial: x²-3x-4=0. Complete all five daily until you solve each in under 60 seconds.

Verify your roots by plugging them back into the original equation.

Vieta's formulas appear in 1-2 ACT Math questions (usually medium difficulty). Knowing them gives you a faster, cleaner path to the answer than the quadratic formula. A student who uses Vieta's formulas solves these questions in 1 minute; a student who relies only on the quadratic formula takes 2-3 minutes. That saved time is valuable buffer on a timed test.

Master this concept in one study session. By test day, Vieta's formulas become your favorite polynomial shortcut.