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A MATH
發問:
Solve the steps very clearly. And Explain.1.Given that k is a non-zero real number. Prove that the quadratic equation kx^2 -x -k = 0 has two unequal real roots.2. If a is a real number and a is not equal to -3, prove that the equation (a^2 +9)x^2 -2(a-3)x+2 = 0 has no real roots.3. Given that k is a non-zero... 顯示更多 Solve the steps very clearly. And Explain. 1.Given that k is a non-zero real number. Prove that the quadratic equation kx^2 -x -k = 0 has two unequal real roots. 2. If a is a real number and a is not equal to -3, prove that the equation (a^2 +9)x^2 -2(a-3)x+2 = 0 has no real roots. 3. Given that k is a non-zero integer. Prove that the quadratic equation x^2 -2kx -3k^2 = 0 has two unequal rational roots.
最佳解答:
(1) kx^2 -x -k = 0 if the quadratic equation has two unequal real roots, discriminant > 0 discriminant = (-1)^2 - 4k(-k) = 1 +4k^2 >0 (because k non zero real number) (2) (a^2 +9)x^2 -2(a-3)x+2 = 0 if the quadratic equation has no real roots, discriminant < 0 discriminant = (2(a-3))^2 - 4(a^2+9) (2) = 4a^2 -24a+36 - 8a^2 -72 = -4a^2 - 24a - 36 =-4 (a^2+6a+9) = -4(a+3)^2 < 0 (because a is not equal to -3) => there are no real roots (discriminant < 0) (3) x^2 -2kx -3k^2 = 0 if the quadratic equation has two unequal real roots, discriminant > 0 discriminant = (-2k)^2 - 4 (1) (-3k^2) = 4k^2 + 12k^2 = 16k^2 > 0 (because k is a non-zero integer) square root of discriminant = 4k, that is also an integer => rational roots
1.Given that k is a non-zero real number. Prove that the quadratic equation kx^2 -x -k = 0 has two unequal real roots. Ans: kx^2 -x -k = 0 Δ= (-1)^2+4(k)(k) =1+4k^2 >0// 2. If a is a real number and a is not equal to -3, prove that the equation (a^2 +9)x^2 -2(a-3)x+2 = 0 has no real roots. Ans: Δ =[2(a-3)]^2-4(a^2+9)(2) =4(a^2-6a+9)-(8a^2+72) =4a^2-24a+36-8a^2-72 =-4a^2-24a-36 = - 4 (a^2+6a+9) = -4(a+3)^2 >0 <---Since to x^2 always bigger than 0 3. Given that k is a non-zero integer. Prove that the quadratic equation x^2 -2kx -3k^2 = 0 has two unequal rational roots. Δ=(2k)^2-4(3k^2) =4k^2-12k^2 =-8k^2 <0 //
A MATH
發問:
Solve the steps very clearly. And Explain.1.Given that k is a non-zero real number. Prove that the quadratic equation kx^2 -x -k = 0 has two unequal real roots.2. If a is a real number and a is not equal to -3, prove that the equation (a^2 +9)x^2 -2(a-3)x+2 = 0 has no real roots.3. Given that k is a non-zero... 顯示更多 Solve the steps very clearly. And Explain. 1.Given that k is a non-zero real number. Prove that the quadratic equation kx^2 -x -k = 0 has two unequal real roots. 2. If a is a real number and a is not equal to -3, prove that the equation (a^2 +9)x^2 -2(a-3)x+2 = 0 has no real roots. 3. Given that k is a non-zero integer. Prove that the quadratic equation x^2 -2kx -3k^2 = 0 has two unequal rational roots.
最佳解答:
(1) kx^2 -x -k = 0 if the quadratic equation has two unequal real roots, discriminant > 0 discriminant = (-1)^2 - 4k(-k) = 1 +4k^2 >0 (because k non zero real number) (2) (a^2 +9)x^2 -2(a-3)x+2 = 0 if the quadratic equation has no real roots, discriminant < 0 discriminant = (2(a-3))^2 - 4(a^2+9) (2) = 4a^2 -24a+36 - 8a^2 -72 = -4a^2 - 24a - 36 =-4 (a^2+6a+9) = -4(a+3)^2 < 0 (because a is not equal to -3) => there are no real roots (discriminant < 0) (3) x^2 -2kx -3k^2 = 0 if the quadratic equation has two unequal real roots, discriminant > 0 discriminant = (-2k)^2 - 4 (1) (-3k^2) = 4k^2 + 12k^2 = 16k^2 > 0 (because k is a non-zero integer) square root of discriminant = 4k, that is also an integer => rational roots
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其他解答:1.Given that k is a non-zero real number. Prove that the quadratic equation kx^2 -x -k = 0 has two unequal real roots. Ans: kx^2 -x -k = 0 Δ= (-1)^2+4(k)(k) =1+4k^2 >0// 2. If a is a real number and a is not equal to -3, prove that the equation (a^2 +9)x^2 -2(a-3)x+2 = 0 has no real roots. Ans: Δ =[2(a-3)]^2-4(a^2+9)(2) =4(a^2-6a+9)-(8a^2+72) =4a^2-24a+36-8a^2-72 =-4a^2-24a-36 = - 4 (a^2+6a+9) = -4(a+3)^2 >0 <---Since to x^2 always bigger than 0 3. Given that k is a non-zero integer. Prove that the quadratic equation x^2 -2kx -3k^2 = 0 has two unequal rational roots. Δ=(2k)^2-4(3k^2) =4k^2-12k^2 =-8k^2 <0 //
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