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我有好多題a-maths 唔識
發問:
1.(a) if a,b and c are real numbers and not all equal, prove that the quadratic equation (c-a)x^2-2(a-b)x=(b-c)=0……..(*) has unequal real roots. (b) if a=1 and b=3, find the discriminant of the equation(*) in (a). hence find the range of values of c so that (*)has unequal real... 顯示更多 1.(a) if a,b and c are real numbers and not all equal, prove that the quadratic equation (c-a)x^2-2(a-b)x=(b-c)=0……..(*) has unequal real roots. (b) if a=1 and b=3, find the discriminant of the equation(*) in (a). hence find the range of values of c so that (*)has unequal real roots. 2.show that the graph of y=3x^2+12x+7+k(x^2-1)cuts the x-axis for any value of k.
1. (a) The equation should be (c-a)x^2-2(a-b)x+(b-c)=0 Discriminant = [2(a - b)]^2 - 4(c - a)(b - c) = 4(a - b)^2 - 4(c - a)(b - c) = 4(a^2 - 2ab + b^2) - 4(bc - ab - c^2 + ac) = 4(a^2 - 2ab + b^2 - bc + ab + c^2 - ac) = 4(a^2 + b^2 + c^2 - ab - bc - ac) = 2[2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ac] = 2[a^2 - 2ab + b^2 + b^2 - 2bc + c^2 + c^2 - 2ca + a^2] = 2[(a - b)^2 + (b - c)^2 + (c - a)^2] > 0 [as a, b, c are unequal, we have (a - b)^2 > 0, (b - c)^2 > 0, (c - a)^2 > 0.] Thus the roots are unequal and real. (b) From (a), discriminant = 2[(1 - 3)^2 + (3 - c)^2 + (c - 1)^2] = 2[4 + c^2 - 6c + 9 + c^2 - 2c + 1] = 2[2c^2 - 8c + 14] = 4[c^2 - 4c + 7] If (*) has unequal roots, then c^2 - 4c + 7 > 0, which is always true as c^2 - 4c + 7 = c^2 - 4c + 4 + 3 = (c - 2)^2 + 3 > 0. Thus c can be any real numbers. 2. y = 3x^2 + 12x + 7 + kx^2 - k = (3 + k)x^2 + 12x + (7 - k) For k =/= -3, discriminant = 12^2 - 4(3 + k)(7 - k) = 144 - 4(21 + 4k - k^2) = 144 - 84 - 16k + 4k^2 = 4k^2 - 16k + 60 = 4(k^2 - 4k + 15) = 4[k^2 - 4k + 4 + 11] = 4(k - 2)^2 + 44 > 0. Thus the graph cuts the x-axis. For k = -3, the equaiton of the graph is y = 12x + 10. Obviously it cuts the x-axis.
其他解答:
我有好多題a-maths 唔識
發問:
1.(a) if a,b and c are real numbers and not all equal, prove that the quadratic equation (c-a)x^2-2(a-b)x=(b-c)=0……..(*) has unequal real roots. (b) if a=1 and b=3, find the discriminant of the equation(*) in (a). hence find the range of values of c so that (*)has unequal real... 顯示更多 1.(a) if a,b and c are real numbers and not all equal, prove that the quadratic equation (c-a)x^2-2(a-b)x=(b-c)=0……..(*) has unequal real roots. (b) if a=1 and b=3, find the discriminant of the equation(*) in (a). hence find the range of values of c so that (*)has unequal real roots. 2.show that the graph of y=3x^2+12x+7+k(x^2-1)cuts the x-axis for any value of k.
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最佳解答:1. (a) The equation should be (c-a)x^2-2(a-b)x+(b-c)=0 Discriminant = [2(a - b)]^2 - 4(c - a)(b - c) = 4(a - b)^2 - 4(c - a)(b - c) = 4(a^2 - 2ab + b^2) - 4(bc - ab - c^2 + ac) = 4(a^2 - 2ab + b^2 - bc + ab + c^2 - ac) = 4(a^2 + b^2 + c^2 - ab - bc - ac) = 2[2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ac] = 2[a^2 - 2ab + b^2 + b^2 - 2bc + c^2 + c^2 - 2ca + a^2] = 2[(a - b)^2 + (b - c)^2 + (c - a)^2] > 0 [as a, b, c are unequal, we have (a - b)^2 > 0, (b - c)^2 > 0, (c - a)^2 > 0.] Thus the roots are unequal and real. (b) From (a), discriminant = 2[(1 - 3)^2 + (3 - c)^2 + (c - 1)^2] = 2[4 + c^2 - 6c + 9 + c^2 - 2c + 1] = 2[2c^2 - 8c + 14] = 4[c^2 - 4c + 7] If (*) has unequal roots, then c^2 - 4c + 7 > 0, which is always true as c^2 - 4c + 7 = c^2 - 4c + 4 + 3 = (c - 2)^2 + 3 > 0. Thus c can be any real numbers. 2. y = 3x^2 + 12x + 7 + kx^2 - k = (3 + k)x^2 + 12x + (7 - k) For k =/= -3, discriminant = 12^2 - 4(3 + k)(7 - k) = 144 - 4(21 + 4k - k^2) = 144 - 84 - 16k + 4k^2 = 4k^2 - 16k + 60 = 4(k^2 - 4k + 15) = 4[k^2 - 4k + 4 + 11] = 4(k - 2)^2 + 44 > 0. Thus the graph cuts the x-axis. For k = -3, the equaiton of the graph is y = 12x + 10. Obviously it cuts the x-axis.
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