Question

Use the quadratic formula to solve the following quadratic equation:

x2−22x+1=15

Answer 1

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \), we first need to set the equation equal to zero. We can do this by subtracting 15 from both sides:

\[
x^2 - 22x + 1 - 15 = 0
\]

This simplifies to:

\[
x^2 - 22x - 14 = 0
\]

Now we can use the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our equation, \( a = 1 \), \( b = -22 \), and \( c = -14 \).

First, we calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = (-22)^2 - 4(1)(-14) = 484 + 56 = 540
\]

Next, we calculate the square root of the discriminant:

\[
\sqrt{540} = \sqrt{36 \times 15} = 6\sqrt{15}
\]

Now we can substitute \( a \), \( b \), and the square root of the discriminant into the quadratic formula:

\[
x = \frac{-(-22) \pm 6\sqrt{15}}{2 \cdot 1}
\]

This simplifies to:

\[
x = \frac{22 \pm 6\sqrt{15}}{2}
\]

Now we can simplify this further by dividing both terms in the numerator by 2:

\[
x = 11 \pm 3\sqrt{15}
\]

Thus, the solutions to the equation \( x^2 - 22x + 1 = 15 \) are:

\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]