체리팟
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Problem #59
Find the value of ∣α−β∣\displaystyle |\alpha -\beta | for the two roots α,β\displaystyle \alpha ,\beta of the quadratic equation 2x2+8x+1=0\displaystyle 2x^{2}+8x+1=0.
Grade Result
Incorrect
1
By the relationship between the roots and coefficients of a quadratic equation:
2
α+β=−4\displaystyle \alpha +\beta =-4, αβ=12\displaystyle \alpha \beta =\frac{1}{2}
3
∴(α−β)2=(α+β)2+4αβ\displaystyle \therefore (\alpha -\beta )^{2}=\left( \alpha +\beta \right)^{2}+4\alpha \beta
😢 Let's try that again and double-check our work. It should be (α−β)2=(α+β)2−4αβ\displaystyle (\alpha -\beta )^{2}=\left( \alpha +\beta \right)^{2}-4\alpha \beta .
4
=(−4)2+4×12\displaystyle =\left( -4 \right)^{2}+4 \times \frac{1}{2}
5
=16+2\displaystyle =16+2
6
=18\displaystyle =18
7
∴∣α−β∣=32\displaystyle \therefore |\alpha -\beta |=3\sqrt{2}
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