Questions: Solve the following equation. For an equation with a real solution, support your answers graphically. x^2 - 9x = x - 22

Solve the following equation. For an equation with a real solution, support your answers graphically.
x^2 - 9x = x - 22
Transcript text: Solve the following equation. For an equation with a real solution, support your answers graphically. \[ x^{2}-9 x=x-22 \]
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Solution

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Solution Steps

To solve the equation \(x^2 - 9x = x - 22\), first rearrange it to form a standard quadratic equation. Then, use the quadratic formula to find the real solutions. Finally, plot the quadratic function to visually confirm the solutions.

Step 1: Move All Terms to One Side

The given equation is:

\[ x^2 - 9x = x - 22 \]

First, move all terms to one side of the equation to set it to zero:

\[ x^2 - 9x - x + 22 = 0 \]

Simplify the equation:

\[ x^2 - 10x + 22 = 0 \]

Step 2: Use the Quadratic Formula

The quadratic formula is given by:

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

For the equation \(x^2 - 10x + 22 = 0\), the coefficients are \(a = 1\), \(b = -10\), and \(c = 22\).

Substitute these values into the quadratic formula:

\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 22}}{2 \cdot 1} \]

Simplify:

\[ x = \frac{10 \pm \sqrt{100 - 88}}{2} \]

\[ x = \frac{10 \pm \sqrt{12}}{2} \]

Step 3: Simplify the Square Root

Simplify \(\sqrt{12}\):

\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]

Substitute back into the equation:

\[ x = \frac{10 \pm 2\sqrt{3}}{2} \]

Step 4: Simplify the Expression

Divide each term by 2:

\[ x = 5 \pm \sqrt{3} \]

Final Answer

The solutions to the equation are:

\[ \boxed{x = 5 + \sqrt{3}} \]

\[ \boxed{x = 5 - \sqrt{3}} \]

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