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A parabola with equation ax^2+bx+c=y has a vertical line of symmetry at X=2 and goes through the two points (1,1)  and (4,-7). The quadratic  ax^2+bx+c has two real roots. The greater root is \(\sqrt n +2 \). What is n?

 Jul 5, 2024
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We can solve this problem by utilizing the properties of the parabola and the given information:

 

Line of Symmetry: Since the line of symmetry is at x = 2, the vertex of the parabola must also be at x = 2.

 

Vertex Form: The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. In this case, the vertex form is y = a(x - 2)^2 + k.

 

Points on the Parabola: We know the parabola passes through the points (1, 1) and (4, -7). We can substitute these points into the vertex form equation to solve for a and k.

 

Substituting (1, 1): 1 = a(1 - 2)^2 + k --> 1 = a + k (Equation 1)

 

Substituting (4, -7): -7 = a(4 - 2)^2 + k --> -7 = 4a + k (Equation 2)

 

Solving for a and k: Subtracting equation 1 from equation 2: -8 = 3a --> a = -8/3. Substitute this value of a back into equation 1: 1 = -8/3 + k --> k = 7/3.

 

Greater Root: We are given that the greater root is sqrt(n) + 2. Since the parabola is symmetric around x = 2, the roots will be equidistant from the vertex (x = 2).

 

Therefore, one root will be less than 2 and the other will be greater than 2. The greater root, sqrt(n) + 2, corresponds to the point where the parabola intersects the x-axis to the right of the vertex.

 

Finding n: Since the parabola intersects the x-axis where y = 0, we can substitute y = 0 and the vertex form equation we derived earlier: 0 = -8/3(x - 2)^2 + 7/3.

 

Solving for x, we can find the greater root. However, we only need the value of n. Since the greater root is squared in the equation, regardless of its positive or negative value, squaring it again will result in a positive value (n).

 

Therefore, we can focus on solving for the value under the square root in the expression sqrt(n) + 2.

 

Simplifying the equation with y = 0: 0 = -8/3(x - 2)^2 + 7/3 --> (x - 2)^2 = 7/8. Taking the square root of both sides (remembering there are positive and negative square roots), we get x - 2 = ±sqrt(7/8).

 

Since we want the value corresponding to the greater root, we take the positive square root: x - 2 = sqrt(7/8) --> x = 2 + sqrt(7/8).

 

Finding n: Now, subtract 2 from both sides to isolate the value under the square root: x - 2 = sqrt(7/8) --> sqrt(n) = sqrt(7/8).

 

Squaring both sides to eliminate the square root: n = 7/8.

 

Therefore, n = 7/8.

 Jul 5, 2024

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