ACT Math: Use Vieta's Formulas to Find Sum and Product of Quadratic Roots

For a quadratic ax^2+bx+c=0 with roots r and s: Sum of roots: r+s=-b/a. Product of roots: r*s=c/a. These relationships let you find roots without solving the quadratic, or verify solutions quickly. Example: x^2-5x+6=0. Sum of roots: -(-5)/1=5. Product: 6/1=6. Roots that sum to 5 and multiply to 6: 2 and 3. (Verify: 2+3=5✓, 2*3=6✓, and (x-2)(x-3)=x^2-5x+6✓) Vieta's formulas are incredibly useful for quickly checking answers or finding roots when factoring is difficult.

Example: 2x^2+3x-2=0. Sum: -3/2. Product: -2/2=-1. You need roots that sum to -3/2 and multiply to -1. Roots are 1/2 and -2. (Verify: 1/2-2=-3/2✓, (1/2)*(-2)=-1✓)

Trap 1: Forgetting to divide by a. If the quadratic is 2x^2+3x-2=0, the sum is not -3 (the coefficient of x); it's -3/2 (the coefficient divided by a). Trap 2: Using Vieta's formulas when you need the actual roots. If a problem asks "what is x?" you can't answer with just the sum and product; you need actual values. Use Vieta's to verify your answer, but use factoring or the quadratic formula to find roots. Remember: a is the coefficient of x^2. Always divide by a when applying Vieta's formulas.

Before you apply Vieta's formulas, identify a, b, and c clearly. Then apply: sum=-b/a, product=c/a. This explicit identification prevents errors.

Problem 1: For x^2-7x+12=0, find the sum and product of roots. Sum: -(-7)/1=7. Product: 12/1=12. Roots that sum to 7 and multiply to 12: 3 and 4. Problem 2: For 3x^2+6x-9=0, find sum and product. Sum: -6/3=-2. Product: -9/3=-3. Roots that sum to -2 and multiply to -3: 1 and -3. (Check: 1-3=-2✓, 1*(-3)=-3✓) Problem 3: If a quadratic has roots 2 and 5, write the quadratic. Sum=7, product=10. Using x^2-(sum)x+product=0: x^2-7x+10=0. Problem 4: For x^2+bx+6=0, one root is 2. Find b and the other root. If one root is 2 and product is 6, the other root is 6/2=3. Sum: 2+3=5=-b, so b=-5. Quadratic: x^2-5x+6=0. All four demonstrate Vieta's formulas in different applications.

Do ten more Vieta's formula problems daily. By test day, using sum and product relationships will be automatic and save you time on quadratic problems.

Vieta's formulas aren't required to solve quadratics, but they're incredibly efficient for verification and quick root-finding when numbers are simple. Students who know and use Vieta's formulas solve certain quadratic problems in seconds while peers spend minutes factoring or using the quadratic formula.

Learn Vieta's formulas this week. Practice applying them to quadratic problems. By test day, you'll use them efficiently to speed up your problem-solving and verify answers instantly.