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If π΄ has coordinates negative seven, π₯ and π΅ has coordinates nine, 14, where π΄π΅ equals four root 17 length units, find all possible values of π₯.
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Letβs begin by thinking about this geometrically.
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Here is point π΅ with coordinates nine, 14.
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It lies in the first quadrant.
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We know point π΄ has an π₯-coordinate of negative seven.
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This means that point π΄ must lie somewhere on the line π₯ equals negative seven.
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For point π΄ to be a distance of four root 17 away from π΅, it could lie in two possible places.
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One is likely to be a little bit above point π΅ in the π¦-direction, and one is likely to be below it.
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Now, at this stage, we donβt necessarily know whether one of these points lies below the π₯-axis.
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Weβre just making a guess.
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But we do know weβre looking for two possible values of π₯.
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So, how do we find these possible values?
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Imagine we have two points on the coordinate plane with coordinates π₯ one, π¦ one and π₯ two, π¦ two.
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The distance between them comes from the Pythagorean theorem.
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Itβs π is equal to the square root of π₯ two minus π₯ one squared plus π¦ two minus π¦ one squared.
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And it doesnβt matter which way round we decide to choose our coordinates.
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Letβs let π₯ one, π¦ one be negative seven, π₯ and π₯ two, π¦ two be nine, 14.
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Then, the distance between these two points is the square root of nine minus negative seven squared plus 14 minus π₯ squared.
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In fact, we can simplify a little bit by recognizing that nine minus negative seven is 16.
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So we get 256, thatβs 16 squared, plus 14 minus π₯ squared inside our root.
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Then, we also note that the length between π΄ and π΅ is given as four root 17 length units.
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So this entire expression must be equal to four root 17.
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We need to solve this equation for π₯.
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So letβs begin by squaring both sides.
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When we do, four root 17 squared is 16 times 17, which is 272.
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And on the right-hand side, we simply have 256 plus 14 minus π₯ squared.
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Then, we have two techniques that we can use to solve this remaining equation.
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We could set it equal to zero and solve as any other quadratic equation.
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Alternatively, letβs see what happens if we subtract 256 from both sides.
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If we subtract 256, our equation becomes 16 equals 14 minus π₯ squared.
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To solve for π₯, weβre going to need to take the square root of both sides.
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But this means we have to take both the positive and negative square root of 16.
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So, we get plus or minus root 16 is equal to 14 minus π₯.
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And of course root 16 is four.
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So, we find that positive or negative four is equal to 14 minus π₯.
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We now need to split this into two separate equations.
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The first will be four equals 14 minus π₯, and the second will be negative four equals 14 minus π₯.
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From both of these equations, weβll subtract 14 from each side.
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That gives us negative 10 equals negative π₯ and negative 18 equals negative π₯, respectively.
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Finally, we multiply through by negative one, and this will give us the values of π₯.
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Thatβs π₯ equals 10 and π₯ equals 18.
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And so we have our two possible values of π₯.
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They are π₯ equals 10 or π₯ equals 18.