Don't Panic! Quadratics and Circle Theorems - MPotW

Given

Find the value of $x$

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The intersecting chords theorem states

| \overset{⇀}{AO} | | \overset{⇀}{BO} | = | \overset{⇀}{CO} | | \overset{⇀}{DO} |

where $A$ , $B$ , $C$ and $D$ are the point where the chords meet the circle, and $O$ is where the chords cross.

From the diagram this simplifies to

(2 x + 5) x = 3

2 x^{2} + 5 x - 3 = 0

2 x^{2} + 6 x - x - 3 = 0

2 x (x + 3) - (x + 3) = 0

(2 x - 1) (x + 3) = 0

Giving $x = \frac{1}{2}$ or $x = - 3$ .

However, the context implies than $x > 0$ , so, $x = \frac{1}{2}$ .

Don't Panic!
Quadratics and Circle Theorems