Find the points where the graphs intersect by forming an equation by making the expressions equal.

${(x - 1)}^{2} = \frac{1}{2} x + 1$

[1]

Expand and simplify the left hand side by expanding $(x - 1) (x - 1) .$

$x^{2} - x - x + 1 = \frac{1}{2} x + 1 x^{2} - 2 x + 1 = \frac{1}{2} x + 1$

[1]

Subtract 1 from both sides.

$x^{2} - 2 x = \frac{1}{2} x$

Subtract $\frac{1}{2} x$ from both sides.

$\begin{array}{rcl} x^{2} - 2 x - \frac{1}{2} x & = & 0 \\ x^{2} - \frac{5}{2} x & = & 0 \end{array}$

[1]

Factorise out the x.

$x (x - \frac{5}{2}) = 0$

Make each factor equal to zero and solve for x.

$\begin{array}{rcl} x & = & 0 \\ x - \frac{5}{2} & = & 0 \Rightarrow x = \frac{5}{2} \end{array}$

[1]

Find the corresponding y-values by substitute the x-values into an equation for one of the graphs.

$\begin{array}{rcl} x & = & 0 \Rightarrow y = {(0 - 1)}^{2} = 1 \\ x & = & \frac{5}{2} \Rightarrow y = {(\frac{5}{2} - 1)}^{2} = \frac{9}{4} \end{array}$

[1]

Write down the coordinates for the two points of intersection.

$A (0, 1) and B (\frac{5}{2}, \frac{9}{4})$

Find the distance between the two points using the formula $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$

$\begin{array}{rcl} d & = & \sqrt{{(\frac{5}{2} - 0)}^{2} + {(\frac{9}{4} - 1)}^{2}} \\ = & \sqrt{\frac{125}{16}} \\ = & \frac{5 \sqrt{5}}{4} \\ = & 2.795084 . . . \end{array}$

[1]

Round to 3 significant figures.

AB = 2.80 [1]

Coordinate Geometry (Cambridge O Level Maths)

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