Solve Solve for : 2x + 22 = x + 1 Solution | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$x : 2 x + 22 = x + 1$$

Solve for x (complex solution)

$x=1+\sqrt{41}i\approx 1+6.403124237i$
$x=-\sqrt{41}i+1\approx 1-6.403124237i$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Multiply both sides of the equation by $2$.

$$xx+44=2x+2$$

Multiply $x$ and $x$ to get $x^{2}$.

$$x^{2}+44=2x+2$$

Subtract $2x$ from both sides.

$$x^{2}+44-2x=2$$

Subtract $2$ from both sides.

$$x^{2}+44-2x-2=0$$

Subtract $2$ from $44$ to get $42$.

$$x^{2}+42-2x=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.

$$x^{2}-2x+42=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $1$ for $a$, $-2$ for $b$, and $42$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 42}}{2}$$

Square $-2$.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 42}}{2}$$

Multiply $-4$ times $42$.

$$x=\frac{-\left(-2\right)±\sqrt{4-168}}{2}$$

Add $4$ to $-168$.

$$x=\frac{-\left(-2\right)±\sqrt{-164}}{2}$$

Take the square root of $-164$.

$$x=\frac{-\left(-2\right)±2\sqrt{41}i}{2}$$

The opposite of $-2$ is $2$.

$$x=\frac{2±2\sqrt{41}i}{2}$$

Now solve the equation $x=\frac{2±2\sqrt{41}i}{2}$ when $±$ is plus. Add $2$ to $2i\sqrt{41}$.

$$x=\frac{2+2\sqrt{41}i}{2}$$

Divide $2+2i\sqrt{41}$ by $2$.

$$x=1+\sqrt{41}i$$

Now solve the equation $x=\frac{2±2\sqrt{41}i}{2}$ when $±$ is minus. Subtract $2i\sqrt{41}$ from $2$.

$$x=\frac{-2\sqrt{41}i+2}{2}$$

Divide $2-2i\sqrt{41}$ by $2$.

$$x=-\sqrt{41}i+1$$

The equation is now solved.

$$x=1+\sqrt{41}i$$ $$x=-\sqrt{41}i+1$$

Steps for Completing the Square

Multiply both sides of the equation by $2$.

$$xx+44=2x+2$$

Multiply $x$ and $x$ to get $x^{2}$.

$$x^{2}+44=2x+2$$

Subtract $2x$ from both sides.

$$x^{2}+44-2x=2$$

Subtract $44$ from both sides.

$$x^{2}-2x=2-44$$

Subtract $44$ from $2$ to get $-42$.

$$x^{2}-2x=-42$$

Divide $-2$, the coefficient of the $x$ term, by $2$ to get $-1$. Then add the square of $-1$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-42+1$$

Add $-42$ to $1$.

$$x^{2}-2x+1=-41$$

Factor $x^{2}-2x+1$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-1\right)^{2}=-41$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-41}$$

Simplify.

$$x-1=\sqrt{41}i$$ $$x-1=-\sqrt{41}i$$

Add $1$ to both sides of the equation.

$$x=1+\sqrt{41}i$$ $$x=-\sqrt{41}i+1$$