Questions: Solve the quadratic equation by completing the square. x^2 - 4x = 6

Solve the quadratic equation by completing the square. x^2 - 4x = 6
Transcript text: Solve the quadratic equation by completing the square. \[ x^{2}-4 x=6 \]
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Solution

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Solution Steps

Step 1: Divide by a

Given the quadratic equation $x^2 - 4x - 6 = 0$, we divide the entire equation by 1 to make the coefficient of $x^2$ equal to 1. This results in $x^2 + \frac{-4}{1}x + \frac{-6}{1} = 0$.

Step 2: Move the constant term to the other side

Rearranging the equation, we get $x^2 + \frac{-4}{1}x = -\frac{-6}{1}$.

Step 3: Complete the square

To complete the square, we add $\left(\frac{-4}{2_1}\right)^2$ to both sides of the equation. This transforms the equation into $x^2 + \frac{-4}{1}x + \left(\frac{-4}{2_1}\right)^2 = \left(\frac{-4}{2*1}\right)^2 - \frac{-6}{1}$.

Step 4: Rewrite the equation as a binomial squared

The equation can now be written as $\left(x + \frac{-4}{2*1}\right)^2 = 4 - \frac{-6}{1}$.

Step 5: Solve for x

Taking the square root of both sides gives us $x + \frac{-4}{2*1} = \pm \sqrt{10}$.

Step 6: Isolate x

Isolating $x$, we find the solutions to be $x = -\frac{-4}{2*1} \pm \sqrt{10}$, which simplifies to $x = 5.16$ and $x = -1.16$.

Final Answer:

The solutions to the quadratic equation $x^2 - 4x - 6 = 0$ are $x = 5.16$ and $x = -1.16$.

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