Evaluate Using the Given Value x^2-2x+7x=1+i

$x^{2} - 2 x + 7 x = 1 + i$

Step 1

Add $- 2 x$ and $7 x$ .

$x^{2} + 5 x = 1 + i$

Step 2

Move all the expressions to the left side of the equation.

Subtract $1$ from both sides of the equation.

$x^{2} + 5 x - 1 = i$

Subtract $i$ from both sides of the equation.

$x^{2} + 5 x - 1 - i = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = 5$ , and $c = - 1 - i$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot (- 1 - i))}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot (- 1 - i)}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot (- 1 - i)}}{2 \cdot 1}$

Apply the distributive property.

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 - 4 (- i)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 - 4 (- i)}}{2 \cdot 1}$

Multiply $- 1$ by $- 4$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 + 4 i}}{2 \cdot 1}$

Add $25$ and $4$ .

$x = \frac{- 5 \pm \sqrt{29 + 4 i}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{29 + 4 i}}{2}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot (- 1 - i)}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot (- 1 - i)}}{2 \cdot 1}$

Apply the distributive property.

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 - 4 (- i)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 - 4 (- i)}}{2 \cdot 1}$

Multiply $- 1$ by $- 4$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 + 4 i}}{2 \cdot 1}$

Add $25$ and $4$ .

$x = \frac{- 5 \pm \sqrt{29 + 4 i}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{29 + 4 i}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 5 + \sqrt{29 + 4 i}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + \sqrt{29 + 4 i}}{2}$

Factor $- 1$ out of $\sqrt{29 + 4 i}$ .

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{29 + 4 i})}{2}$

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{29 + 4 i})$ .

$x = \frac{- 1 (5 - \sqrt{29 + 4 i})}{2}$

Move the negative in front of the fraction.

$x = - \frac{5 - \sqrt{29 + 4 i}}{2}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot (- 1 - i)}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot (- 1 - i)}}{2 \cdot 1}$

Apply the distributive property.

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 - 4 (- i)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 - 4 (- i)}}{2 \cdot 1}$

Multiply $- 1$ by $- 4$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 + 4 i}}{2 \cdot 1}$

Add $25$ and $4$ .

$x = \frac{- 5 \pm \sqrt{29 + 4 i}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{29 + 4 i}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 5 - \sqrt{29 + 4 i}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - \sqrt{29 + 4 i}}{2}$

Factor $- 1$ out of $- \sqrt{29 + 4 i}$ .

$x = \frac{- 1 \cdot 5 - (\sqrt{29 + 4 i})}{2}$

Factor $- 1$ out of $- 1 (5) - (\sqrt{29 + 4 i})$ .

$x = \frac{- 1 (5 + \sqrt{29 + 4 i})}{2}$

Move the negative in front of the fraction.

$x = - \frac{5 + \sqrt{29 + 4 i}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{29 + 4 i}}{2}, - \frac{5 + \sqrt{29 + 4 i}}{2}$

Evaluate Using the Given Value x^2-2x+7x=1+i

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