Let's go through the process of solving the quadratic equation [tex]\( 5x^2 + 20x - 7 = 0 \)[/tex] by completing the square step-by-step. Based on the given information, we'll identify the correct steps Sergey could use.
1. Step 1: Isolate the quadratic and linear terms
First, we start by moving the constant term to the other side:
[tex]\[
5x^2 + 20x = 7
\][/tex]
2. Step 2: Normalize the coefficient of [tex]\( x^2 \)[/tex]
The equation is already in the form with a normalized [tex]\( x^2 \)[/tex] term, so no scaling is needed.
3. Step 3: Complete the square
We form a perfect square trinomial from the quadratic and linear terms inside the equation:
[tex]\[
x^2 + 4x + 4
\][/tex]
This can be written as:
[tex]\[
(x + 2)^2
\][/tex]
4. Step 4: Add and subtract the square term
To balance the equation, we multiply back with the coefficient of [tex]\( x^2 \)[/tex] and add the necessary constants:
[tex]\[
5(x^2 + 4x + 4) = 7 + 20
\][/tex]
Simplifying, we get:
[tex]\[
5(x^2 + 4x + 4) = 27
\][/tex]
5. Step 5: Solve the equation
Now we can solve the simplified equation:
[tex]\[
(x + 2)^2 = \frac{27}{5}
\][/tex]
Taking the square root on both sides:
[tex]\[
x + 2 = \pm \sqrt{\frac{27}{5}}
\][/tex]
Therefore, the correct steps Sergey could use are:
1. [tex]\( 5(x^2 + 4x) = 7 \)[/tex]
2. [tex]\( 5(x^2 + 4x + 4) = 7 + 20 \)[/tex]
3. [tex]\( x + 2 = \pm \sqrt{\frac{27}{5}} \)[/tex]
These steps lead us to the solution:
- [tex]\( 5(x^2 + 4x) = 7 \)[/tex]
- [tex]\( 5(x^2 + 4x + 4) = 7 + 20 \)[/tex]
- [tex]\( x + 2 = \pm \sqrt{\frac{27}{5}} \)[/tex]
are the correct steps. These are:
- [tex]\( 5(x^2+4x+4)=7+20 \)[/tex]
- [tex]\( x+2= \pm \sqrt{\frac{27}{5}} \)[/tex]
- [tex]\( 5\left(x^2+4 x\right)=7 \)[/tex]
