Question

Solve by completing the square:
x2 + 3x – 9 = 0

Answer 1

`x=(-3\pm3√(5))/(2)`

Explanation:

Given equation:

`x^2+3x-9=0`

Completing the square

Move the constant to the right side by adding 9 to both sides:

`\implies x^2+3x=9`

Add the square of half the coefficient of x to both sides:

`\implies x^2+3x+\left((3)/(2)\right)^2=9+\left((3)/(2)\right)^2`

`\implies x^2+3x+(9)/(4)=(45)/(4)`

Factor the trinomial on the left side:

`\implies \left(x+(3)/(2)\right)^2=(45)/(4)`

Square root both sides:

`\implies x+(3)/(2)=\pm\sqrt{(45)/(4)}`

`\implies x+(3)/(2)=\pm(√(45))/(√(4))}`

`\implies x+(3)/(2)=\pm(√(9 \cdot 5))/(2)`

`\implies x+(3)/(2)=\pm(√(9) √(5))/(2)`

`\implies x+(3)/(2)=\pm(3√(5))/(2)`

Subtract 3/2 from both sides:

`\implies x=\pm(3√(5))/(2)-(3)/(2)`

`\implies x=(-3\pm3√(5))/(2)`