Question

Solve by completing the square:
`{x}^(2) + 10x - 24 = 0`

Answer 1

Let's solve the given equation using completing the square:

`\text{ x}^2\text{ + 10x - 24 = 0}`

Step 1: Keep x terms on the left and move the constant to the right side by adding it on both sides.

`\text{ x}^2\text{ + 10x - 24 = 0}`
`\text{ x}^2\text{ + 10x - 24 + 24 = 0 + 24}`
`\text{ x}^2\text{ + 10x = 24}`

Step 2: Take half of the x term and square it.

`\text{ x term = 10x}`
`((b)/(2))^2\text{ = (}(10)/(2))^2=5^2\text{ = 25}`

Step 3: Add the result to both sides.

`\text{ x}^2\text{ + 10x = 24}`
`\text{ (x}^2\text{ + 10x + 25) = 24 + 25}`

Step 4: Rewrite the perfect square on the left and combine terms on the right.

`\text{ (x}^2\text{ + 10x + 25) = 24 + 25}`
`(x+5)^2\text{ = 49}`

Step 5: Square root of both sides.

`√((x+5)^2)\text{ = }\sqrt{\text{49}}`
`\text{ x + 5 = }\pm\text{ 7}`
`\text{ x = }\pm\text{ 7 - 5}`

Step 6: Solve for x.

`x_1\text{ = 7 - 5 = 2}`
`x_2\text{ = -7 - 5 = -12}`

Therefore, there are two solutions: x = 2 and -12.