Question

Sqrt (x^2-4x+3)=sqrt (3x+a) Найти значения a, при котором уравнение имеет один корень

Answer 1

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Answer 2

To find the values of "a" for which the equation has a single root, we need to set the discriminant of each square root expression to zero. This is because when the discriminant is zero, the square root expression will have a double root, resulting in a single solution for the entire equation.

The equation is: √(x^2 - 4x + 3) = √(3x + a)

Step 1: Set the discriminant of the first square root expression to zero: x^2 - 4x + 3 = 0

Step 2: Solve for "x" using the quadratic formula: x = [4 ± √(4^2 - 4 * 1 * 3)] / 2 x = [4 ± √(16 - 12)] / 2 x = [4 ± √4] / 2 x = [4 ± 2] / 2

So, the two possible values of "x" are: x₁ = (4 + 2) / 2 = 3 x₂ = (4 - 2) / 2 = 1

Step 3: For the equation to have a single root, both square root expressions on the left and right sides of the equation must yield the same value for "x". So, we need to substitute both "x₁" and "x₂" into the right side of the equation and find the corresponding values of "a" for each case.

For x = 3: √(3 * 3 + a) = √(9 + a) 3 + a = 9 + a a = 9 + a - 3 a = 6

For x = 1: √(3 * 1 + a) = √(3 + a) 1 + a = √(3 + a)² 1 + a = 3 + a 1 = 3 (Not possible, as the equation does not hold true for x = 1)

Thus, the equation has a single root when "a" is equal to 6. For any other value of "a," the equation will either have two distinct roots or no real roots.