How do I solve the following problem? The lesson is on turning quadratics into vertex form and/or completing the square so I assume that is part of the process.Amanda and Dolvin leave their house...

How do I solve the following problem? The lesson is on turning quadratics into vertex form and/or completing the square so I assume that is part of the process.

Amanda and Dolvin leave their house at the same time. Amanda walks south and Dolvin bikes east. Half an hour later they are 5.5 miles away from each other and Dolvin has covered three miles more than the distance that Amanda covered. How far did Amanda walk and how far did Dolvin bike?

Asked on by catd1115

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should come up with the following notations:x expresses the distance covered by  Amanda half an hour later, x + 3 expresses the distance covered by Dolvin half an hour later and 5.5 is the distance between Amanda and Dolvin, half an hour later.

You need to remember that east direction is perpendicular to south direction, hence joining the point where Amanda arrived half and hour later to the point where Dolvin arrived half an hour later yields the hypotenuse of a right triangle whose legs are distances covered by Amanda and Dolvin.

Hence, you should use Pythagorean theorem such that:

`(5.5)^2 = x^2 + (x + 3)^2`

`30.25 = x^2 + x^2 + 6x + 9`

You need to move all terms to left side:

`-2x^2 - 6x + 21.25 = 0 =gt 2x^2 + 6x- 21.25 = 0`

You need to divide by 2 both sides such that:

`x^2 + 3x - 10.625 = 0`

You need to use quadratic formula such that:

`x_(1,2) = (-3 +- sqrt(9 + 42.5))/2 =gt x_(1,2) = (-3 +- sqrt51.5)/2`

`x_1 = (-3+7.1)/2 =gt x_1 = 2.05`

`x_2 = -5.05`

You need to keep the nonnegative value for x, hence, half an hour later, Amanda covered 2.05 miles and Dolvin covered 5.05 miles.

You need to conver the standard form of quadratic equation `x^2 + 3x - 10.625 = 0`  into vertex form, hence, you need to complete the square x^2 + 3x  by the positive constant term  `9/4` . You need to subtract the same constant  term `9/4`  to keep the equation balanced such that:

`(x^2 + 3x + 9/4) - 9/4 - 10.625 = 0`

`(x + 3/2)^2 - 12.875 = 0`

Hence, the vertex form of quadratic that helps you to find the distances of 2.05 miles and 5.05 miles, covered by Amanda and Dolvin, is `(x + 3/2)^2 - 12.875 = 0` .

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