Question

Work out x when the area is root 4 2^2 and angle C is 45 and ab are (x+2)(2x-3)

Answer 1

• x = 3.08

Explanation:

Area formula:

• A = 1/2bh

The height is:

• h = (x + 2)sin 45° = (x + 2) ×√2/2

Substitute the value for area and set equation:

• 4√2 = 1/2 × (2x - 3) × √2(x + 2)/2
• (2x - 3)(x + 2) = 16
• 2x² + x - 6 = 16
• 2x² + x - 22 = 0
• D = 1 + 4*2*22 = 177
• x = (-1 + √177)/4 = 3.08 (rounded)

Note the second root ignored as negative

Answer 2

x = 3.08 (3 s.f.)

Explanation:

Area of a Triangle (using sine)

`\sf Area=(1)/(2)ac \sin B`

where:

• a and c are adjacent sides
• B is the included angle

Given:

• Area = 4√2 m²
• a = (2x - 3) m
• c = (x + 2) m
• B = 45°

Substitute the given values into the formula:

`\implies 4√(2)=(1)/(2)(2x-3)(x+2) \sin 45^(\circ)`

`\implies 8√(2)=(2x-3)(x+2) \sin 45^(\circ)`

`\implies 8√(2)=(2x-3)(x+2) \left((√(2))/(2)\right)`

`\implies 8√(2)\left((2)/(√(2))\right)=(2x-3)(x+2)`

`\implies (2x-3)(x+2) =16`

`\implies 2x^2+4x-3x-6=16`

`\implies 2x^2+x-22=0`

Use the Quadratic Formula to solve for x:

`x=(-b \pm √(b^2-4ac) )/(2a)\quad\textsf{when }\:ax^2+bx+c=0`

`\implies a=2, \quad b=1, \quad c=-22`

`\implies x=(-1 \pm √(1^2-4(2)(-22)) )/(2(2))`

`\implies x=(-1 \pm √(177))/(4)`

`\implies x=3.08, -3.58\:\: \sf (3 \:s.f.)`

As distance is positive:

`\implies x=3.08 \:\:(3 \sf \:s.f.) \:\textsf{only}`