If the roots of the equation [tex]\left(a^2+b^2\right) x^2 - 2(a c + b d) x + \left(c^2+d^2\right) = 0[/tex] are equal, then prove that [tex]\frac{a}{b} = \frac{c}{d}[/tex].

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01-08-2024
Mathematics

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If the roots of the equation [tex]\left(a^2+b^2\right) x^2 - 2(a c + b d) x + \left(c^2+d^2\right) = 0[/tex] are equal, then prove that [tex]\frac{a}{b} = \frac{c}{d}[/tex].

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-08-01T08:45:30-04:00

To prove that [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex] when the roots of the equation [tex]\((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\)[/tex] are equal, follow these steps:

1. Understand the general form of the quadratic equation:
The given equation can be written in the standard quadratic form [tex]\(Ax^2 + Bx + C = 0\)[/tex], where:
[tex]\[ A = a^2 + b^2, \quad B = -2(ac + bd), \quad C = c^2 + d^2. \][/tex]

2. Condition for equal roots:
To have equal roots for the quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex], the discriminant [tex]\(\Delta\)[/tex] must be zero. The discriminant of a quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex] is given by:
[tex]\[ \Delta = B^2 - 4AC. \][/tex]
Setting the discriminant to zero:
[tex]\[ B^2 - 4AC = 0. \][/tex]

3. Substitute [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
Substitute the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] into the discriminant equation:
[tex]\[ (-2(ac + bd))^2 - 4(a^2 + b^2)(c^2 + d^2) = 0. \][/tex]

4. Simplify the equation:
Simplify the expression:
[tex]\[ 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2) = 0. \][/tex]
Divide through by 4 to simplify further:
[tex]\[ (ac + bd)^2 - (a^2 + b^2)(c^2 + d^2) = 0. \][/tex]
Expanding both sides:
[tex]\[ (ac + bd)^2 = (a^2 + b^2)(c^2 + d^2). \][/tex]
Expand [tex]\((ac + bd)^2\)[/tex]:
[tex]\[ a^2c^2 + 2abcd + b^2d^2 = a^2c^2 + b^2c^2 + a^2d^2 + b^2d^2. \][/tex]

5. Compare both sides:
Comparing the terms on both sides of the equation:
[tex]\[ 2abcd = b^2c^2 + a^2d^2. \][/tex]
Rearrange the terms to isolate [tex]\(abcd\)[/tex]:
[tex]\[ a^2d^2 + b^2c^2 - 2abcd = 0. \][/tex]
Recognize this as a perfect square:
[tex]\[ (ad - bc)^2 = 0. \][/tex]
Therefore:
[tex]\[ ad - bc = 0. \][/tex]

6. Conclude the proof:
From [tex]\(ad - bc = 0\)[/tex], we obtain:
[tex]\[ ad = bc. \][/tex]
Finally, dividing both sides by [tex]\(bd\)[/tex]:
[tex]\[ \frac{a}{b} = \frac{c}{d}. \][/tex]

Thus, we have proven that if the roots of the equation [tex]\((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\)[/tex] are equal, then [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex].