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}$$(\frac{b}{2})^2\text{ = (}\frac{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.

$\sqrt{(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.