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A parabola with equation \(y=ax^2+bx+c\) 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 \(n?\)

 May 19, 2023
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The vertical line of symmetry at x=2 means that the parabola is symmetric around the line x=2. This means that the midpoint of the segment connecting the points (1,1) and (4,-7) is the vertex of the parabola. The midpoint of the segment is (2.5,-3). The vertex is also the average of the roots of the quadratic, so the average of the roots is -3. Since the greater root is \sqrt{n}+2, the lesser root is -3-(\sqrt{n}+2)=-\sqrt{n}-4. The sum of the roots is 0, so \sqrt{n}-4-(-\sqrt{n}-4)=0. This simplifies to 2\sqrt{n}=8, so \sqrt{n}=4. Therefore, n=16.

Here are the steps in more detail:

The vertical line of symmetry at x=2 means that the parabola is symmetric around the line x=2. This means that the midpoint of the segment connecting the points (1,1) and (4,-7) is the vertex of the parabola.

The midpoint of the segment is (2.5,-3).

The vertex is also the average of the roots of the quadratic, so the average of the roots is -3.

Since the greater root is \sqrt{n}+2, the lesser root is -3-(\sqrt{n}+2)=-\sqrt{n}-4.

The sum of the roots is 0, so \sqrt{n}-4-(-\sqrt{n}-4)=0.

This simplifies to 2\sqrt{n}=8, so \sqrt{n}=4.

Therefore, n=16.

 May 19, 2023

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