Question

Sole the quadratic equation by completing the square. x^2+10x+15=0Choose the appropriate form and fill in the blanks with the correct numbers. Then solve the equation. If there’s more than one solution, separate them with commas.

Answer 1

In the function:

`y=x^2+10x+15`

the coefficients are:

a = 1

b = 10

c = 15

Then, b/2 = 10/2 = 5. Computing the square of x and b/2, we get:

`(x+5)^2=x^2+2\cdot x\cdot5+5^2=x^2+10x+25`

We can see that the first two terms coincide with the previous function, then we need to add and subtract 25 to that function to complete the square, as follows:

`\begin{gathered} y=x^2+10+15 \\ y=x^2+10+15+25-25 \\ y=(x^2+10+25)+(15-25) \\ y=(x+5)^2-10 \end{gathered}`

Solving the equation:

`\begin{gathered} (x+5)^2-10=0 \\ (x+5)^2=0+10 \\ (x+5)^2=10 \\ x+5=\sqrt[]{10} \\ This\text{ equation has 2 solutions:} \\ x_1=\sqrt[]{10}-5 \\ Or \\ x_2=-\sqrt[]{10}-5 \end{gathered}`