Asked by a mr
factoring trinomials with leading coefficient 6x to the power of 2 -10x+50. 4x to the power of 2 -4x-63
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GPT-5 mini
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Here are the factorizations and brief work:
1) 6x^2 − 10x + 50
- Factor out GCF 2: 2(3x^2 − 5x + 25).
- The quadratic 3x^2 − 5x + 25 has discriminant Δ = (−5)^2 − 4·3·25 = 25 − 300 = −275 < 0, so it has no real (or rational) linear factors.
- Final (over integers/rationals): 2(3x^2 − 5x + 25).
- Over the complex numbers you can write the linear factors using the roots x = (5 ± 5i√11)/6, so 6x^2 − 10x + 50 = 6(x − (5+5i√11)/6)(x − (5−5i√11)/6).
2) 4x^2 − 4x − 63
- Use the ac-method: a·c = 4·(−63) = −252. Numbers that multiply to −252 and add to −4 are 14 and −18.
- Split middle term: 4x^2 + 14x − 18x − 63 = 2x(2x+7) − 9(2x+7) = (2x+7)(2x−9).
- Final: (2x + 7)(2x − 9).
1) 6x^2 − 10x + 50
- Factor out GCF 2: 2(3x^2 − 5x + 25).
- The quadratic 3x^2 − 5x + 25 has discriminant Δ = (−5)^2 − 4·3·25 = 25 − 300 = −275 < 0, so it has no real (or rational) linear factors.
- Final (over integers/rationals): 2(3x^2 − 5x + 25).
- Over the complex numbers you can write the linear factors using the roots x = (5 ± 5i√11)/6, so 6x^2 − 10x + 50 = 6(x − (5+5i√11)/6)(x − (5−5i√11)/6).
2) 4x^2 − 4x − 63
- Use the ac-method: a·c = 4·(−63) = −252. Numbers that multiply to −252 and add to −4 are 14 and −18.
- Split middle term: 4x^2 + 14x − 18x − 63 = 2x(2x+7) − 9(2x+7) = (2x+7)(2x−9).
- Final: (2x + 7)(2x − 9).
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