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Anna was asked to solve the following quadratic equation by completing the square:

[tex]\[ x^2 + 6x - 7 = 0 \][/tex]

Her first step was to add 7 to both sides:

[tex]\[ x^2 + 6x = 7 \][/tex]

Next, she adds [tex]\( c \)[/tex] to both sides of the equation to create a perfect square trinomial on the left:

[tex]\[ x^2 + 6x + c = 7 + c \][/tex]

What is the value of [tex]\( c \)[/tex]?

Answer: [tex]\( c = 9 \)[/tex]



Answer :

GinnyAnswer
To solve the quadratic equation [tex]\( x^2 + 6x - 7 = 0 \)[/tex] by completing the square, we need to follow these steps:

1. Start with the given equation:
[tex]\[ x^2 + 6x - 7 = 0 \][/tex]

2. Isolate the terms containing [tex]\( x \)[/tex] on one side of the equation:
Add 7 to both sides to balance the equation:
[tex]\[ x^2 + 6x = 7 \][/tex]

3. Find the value [tex]\( c \)[/tex] that completes the square:
To complete the square, we need to make the left side of the equation a perfect square trinomial. The perfect square trinomial takes the form [tex]\((x + b)^2\)[/tex], where [tex]\( b \)[/tex] is half the coefficient of [tex]\( x \)[/tex].

The coefficient of [tex]\( x \)[/tex] is 6. We take half of this coefficient:
[tex]\[ \frac{6}{2} = 3 \][/tex]

Next, square this result:
[tex]\[ 3^2 = 9 \][/tex]

4. Add [tex]\( c \)[/tex] to both sides of the equation:
The value of [tex]\( c \)[/tex] is 9, so add it to both sides:
[tex]\[ x^2 + 6x + 9 = 7 + 9 \][/tex]

5. Verify the value of [tex]\( c \)[/tex]:
It is clear that adding 9 completes the square.

Therefore, the value of [tex]\( c \)[/tex] is:
[tex]\[ c = 9 \][/tex]

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