Question

Use the following construction to solve the equation x² + bx = c. Please include diagram.

Construction:

1) Create a segment AB of length √c

2) Draw a line AC perpendicular to AB w/ a length of b/2

3) Draw a circle w/ radius AC & center C

4) Let D be the point where BC hits the circle

5) BE is the solution x we want to solve

Prove that x² + bx = c

Answer 1

The construction method creates a right triangle that satisfies the equation x² + bx = c through geometric relationships and the Pythagorean Theorem, proving that the length BE is the solution to the quadratic equation.

Step-by-step explanation:

The equation x² + bx = c can be solved using a geometric construction. The following steps detail this construction technique:

1. Create a segment AB of length √c.
2. Draw a line AC perpendicular to AB with a length of b/2.
3. Draw a circle with radius AC and center C.
4. Let D be the point where line BC hits the circle.
5. Segment BE represents the solution x we want to solve.

To prove that BE is the solution to the equation, we apply the Pythagorean Theorem (a² + b² = c²) to the right triangle formed by AB, BE, and AE (where AE is the extension of AB). Since AB = √c, and AC (which will be part of BE) is equal to b/2, it follows that:

(b/2)² + x² = (√c)²

Rearranging this, we get:

x² + (b²/4) = c

And, if we double the value of BE in the equation to account for the full length of AC, we get the original equation:

x² + bx = c

This proves that the length BE, as found through the construction, is indeed the solution to the equation.