How it works
Multiplying out (x + m)(x + n) gives x² + (m + n)x + mn. So factoring x² + 13x + 36 means finding numbers with product 36 and sum 13: 4 and 9, giving (x + 4)(x + 9).
The roots are the values that make a factor zero, so here they are −4 and −9. This gives Vieta's formulas for x² + bx + c = 0 with roots r and s:
r + s = −b rs = c
Vieta lets you answer questions about the roots (their sum, product, reciprocals or difference) without finding them. Signs are the usual trap: if c > 0 and b < 0, both roots are positive.
Factoring Monic Quadratics practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
Factor x²+13x+36. What is the larger constant in the factors?
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Answer: 9
We need two numbers with product 36 and sum 13: 4 and 9. So x² + 13x + 36 = (x + 4)(x + 9), and the larger constant is 9.
x²+bx+14 = 0 has roots −2 and −7. What is b?
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Answer: 9
The sum of the roots is −b: −2 + (−7) = −9, so b = 9. Check: (x + 2)(x + 7) = x² + 9x + 14.
x²+bx+16 = 0 has integer roots differing by 6. If b is negative, what is b?
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Answer: −10
The roots multiply to 16 and differ by 6, so they are 2 and 8 or −2 and −8. Since b = −(sum of roots) is negative, the sum is positive: the roots are 2 and 8. So b = −10. Check: (x − 2)(x − 8) = x² − 10x + 16.
If r+s = 9 and rs = 14, what is 1/r+1/s?
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Answer: 9/14
Combine the fractions: 1/r + 1/s = (r + s) ÷ rs = 9/14.
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