Question

Please help me with this problem:Solve the quadratic equation x^2 + 2x = 35 using two different methodsAnswer:Method 1:Method 2:

Answer 1

Step-by-step explanation

We must solve by two different methods the following equation:

`x^2+2x=35.`

Method 1: Completing the square

1) We rewrite the equation above as:

`x^2+2\cdot1\cdot x=35.`

2) Now, we add 1² on both sides of the equation:

`(x^2+2\cdot1\cdot x+1^2)=35+1^2=36.`

3) We see that the left and right sides can be written as squares:

`(x+1)^2=6^2.`

4) Taking the square root on both sides, we get two solutions:

`\begin{gathered} x+1=+6\Rightarrow x_1=6-1=5, \\ x+1=-6\Rightarrow x_2=-6-1=-7. \end{gathered}`

Method 2: Using the quadratic formula

1) We rewrite the equation above as:

`x^2+2x-35=0.`

2) We identify a quadratic equation:

`a\cdot x^2+b\cdot x+c=0.`

With coefficients:

• a = 1,

,

• b = 2,

,

• c = -35.

The roots of this equation are given by:

`\begin{gathered} x_1=(-b+√(b^2-4ac))/(2a), \\ x_2=(-b-√(b^2-4ac))/(2a). \end{gathered}`

3) Replacing the coefficients of the quadratic equation in the formula above, we get:

`\begin{gathered} x_1=(-2+√(2^2-4\cdot1\cdot(-35)))/(2\cdot1)=(-2+√(144))/(2)=(-2+12)/(2)=(10)/(2)=5, \\ x_2=(-2-√(2^2-4\cdot1\cdot(-35)))/(2\cdot1)=(-2-√(144))/(2)=(-2-12)/(2)=-(14)/(2)=-7. \end{gathered}`Answer

The roots of the polynomial are:

• x₁ = 5

,

• x₂ = -7