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QuestionRewrite the equation x2−14x−1x^{2}-14 x-1 as (x+a)2+b(x+ a)^2 + b with integers aa and bb.

Studdy Solution

STEP 1

Assumptions1. The given quadratic equation is x−14x−1x^{}-14 x-1 . We need to rewrite this equation in the form (x+a)+b(x+ a)^ + b where aa and bb are integers.

STEP 1

Assumptions1. The given quadratic equation is x−14x−1x^{}-14 x-1. . We need to rewrite it in the form (x+a)+b(x+ a)^ + b where aa and bb are integers.

STEP 2

The general form of a quadratic equation is ax2+bx+cax^{2}+bx+c. To rewrite it in the form (x+a)2+b(x+ a)^2 + b, we need to complete the square.

STEP 2

The general form of a quadratic equation is ax2+bx+cax^2 + bx + c. We can rewrite this in the form (x+a)2+b(x+ a)^2 + b by completing the square. The formula for completing the square is (x+b2a)2−(b2a)2+c(x + \frac{b}{2a})^2 - (\frac{b}{2a})^2 + c.

STEP 3

The first step in completing the square is to identify the coefficient of the xx term, which in this case is −14-14.

STEP 3

In our equation, a=1a =1 and b=−14b = -14. So, we can substitute these values into the formula.
x2−14x−1=(x+−142(1))2−(−142(1))2−1x^2 -14x -1 = \left(x + \frac{-14}{2(1)}\right)^2 - \left(\frac{-14}{2(1)}\right)^2 -1

STEP 4

Next, we divide this coefficient by 22 and square the result to get the value that will complete the square. This will give us the value of aa.
a=(−142)2a = \left(\frac{-14}{2}\right)^2

STEP 4

implify the equation.
x2−14x−1=(x−7)2−72−1x^2 -14x -1 = \left(x -7\right)^2 -7^2 -1

STEP 5

Calculate the value of aa.
a=(−142)2=49a = \left(\frac{-14}{2}\right)^2 =49

STEP 5

Calculate the square of7 and simplify the equation.
x2−14x−1=(x−7)2−49−1x^2 -14x -1 = \left(x -7\right)^2 -49 -1

STEP 6

Now that we have the value of aa, we substitute this into the equation to complete the square.
x2−14x−1=(x−)2−49−1x^{2}-14 x-1 = (x-)^2 -49 -1

STEP 6

Combine like terms.
x2−14x−1=(x−)2−50x^2 -14x -1 = \left(x -\right)^2 -50

STEP 7

implify the equation to find the value of bb.
x2−14x−1=(x−7)2−50x^{2}-14 x-1 = (x-7)^2 -50So, the quadratic equation x2−14x−1x^{2}-14 x-1 can be rewritten in the form (x+a)2+b(x+ a)^2 + b as (x−7)2−50(x-7)^2 -50 where a=−7a = -7 and b=−50b = -50.

STEP 7

Now, we have the equation in the form (x+a)2+b(x+ a)^2 + b. Here, a=−7a = -7 and b=−50b = -50.
So, the quadratic equation x2−14x−1x^{2}-14 x-1 can be rewritten as (x−7)2−50(x -7)^2 -50.

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