Solve (1)/(x-1)+(2)/(2x-1)=3 | 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

$$\frac{ 1 }{ x-1 } + \frac{ 2 }{ 2x-1 } =3$$

Solve for x

$x=\frac{2}{3}\approx 0.666666667$
$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to any of the values $\frac{1}{2},1$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-1\right)\left(2x-1\right)$, the least common multiple of $x-1,2x-1$.

$$2x-1+\left(x-1\right)\times 2=3\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply $x-1$ by $2$.

$$2x-1+2x-2=3\left(x-1\right)\left(2x-1\right)$$

Combine $2x$ and $2x$ to get $4x$.

$$4x-1-2=3\left(x-1\right)\left(2x-1\right)$$

Subtract $2$ from $-1$ to get $-3$.

$$4x-3=3\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply $3$ by $x-1$.

$$4x-3=\left(3x-3\right)\left(2x-1\right)$$

Use the distributive property to multiply $3x-3$ by $2x-1$ and combine like terms.

$$4x-3=6x^{2}-9x+3$$

Subtract $6x^{2}$ from both sides.

$$4x-3-6x^{2}=-9x+3$$

Add $9x$ to both sides.

$$4x-3-6x^{2}+9x=3$$

Combine $4x$ and $9x$ to get $13x$.

$$13x-3-6x^{2}=3$$

Subtract $3$ from both sides.

$$13x-3-6x^{2}-3=0$$

Subtract $3$ from $-3$ to get $-6$.

$$13x-6-6x^{2}=0$$

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.

$$-6x^{2}+13x-6=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-6$ for $a$, $13$ for $b$, and $-6$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-6\right)\left(-6\right)}}{2\left(-6\right)}$$

Square $13$.

$$x=\frac{-13±\sqrt{169-4\left(-6\right)\left(-6\right)}}{2\left(-6\right)}$$

Multiply $-4$ times $-6$.

$$x=\frac{-13±\sqrt{169+24\left(-6\right)}}{2\left(-6\right)}$$

Multiply $24$ times $-6$.

$$x=\frac{-13±\sqrt{169-144}}{2\left(-6\right)}$$

Add $169$ to $-144$.

$$x=\frac{-13±\sqrt{25}}{2\left(-6\right)}$$

Take the square root of $25$.

$$x=\frac{-13±5}{2\left(-6\right)}$$

Multiply $2$ times $-6$.

$$x=\frac{-13±5}{-12}$$

Now solve the equation $x=\frac{-13±5}{-12}$ when $±$ is plus. Add $-13$ to $5$.

$$x=-\frac{8}{-12}$$

Reduce the fraction $\frac{-8}{-12}$ to lowest terms by extracting and canceling out $4$.

$$x=\frac{2}{3}$$

Now solve the equation $x=\frac{-13±5}{-12}$ when $±$ is minus. Subtract $5$ from $-13$.

$$x=-\frac{18}{-12}$$

Reduce the fraction $\frac{-18}{-12}$ to lowest terms by extracting and canceling out $6$.

$$x=\frac{3}{2}$$

The equation is now solved.

$$x=\frac{2}{3}$$ $$x=\frac{3}{2}$$

Steps for Completing the Square

Variable $x$ cannot be equal to any of the values $\frac{1}{2},1$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-1\right)\left(2x-1\right)$, the least common multiple of $x-1,2x-1$.

$$2x-1+\left(x-1\right)\times 2=3\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply $x-1$ by $2$.

$$2x-1+2x-2=3\left(x-1\right)\left(2x-1\right)$$

Combine $2x$ and $2x$ to get $4x$.

$$4x-1-2=3\left(x-1\right)\left(2x-1\right)$$

Subtract $2$ from $-1$ to get $-3$.

$$4x-3=3\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply $3$ by $x-1$.

$$4x-3=\left(3x-3\right)\left(2x-1\right)$$

Use the distributive property to multiply $3x-3$ by $2x-1$ and combine like terms.

$$4x-3=6x^{2}-9x+3$$

Subtract $6x^{2}$ from both sides.

$$4x-3-6x^{2}=-9x+3$$

Add $9x$ to both sides.

$$4x-3-6x^{2}+9x=3$$

Combine $4x$ and $9x$ to get $13x$.

$$13x-3-6x^{2}=3$$

Add $3$ to both sides.

$$13x-6x^{2}=3+3$$

Add $3$ and $3$ to get $6$.

$$13x-6x^{2}=6$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$-6x^{2}+13x=6$$

Divide both sides by $-6$.

$$\frac{-6x^{2}+13x}{-6}=\frac{6}{-6}$$

Dividing by $-6$ undoes the multiplication by $-6$.

$$x^{2}+\frac{13}{-6}x=\frac{6}{-6}$$

Divide $13$ by $-6$.

$$x^{2}-\frac{13}{6}x=\frac{6}{-6}$$

Divide $6$ by $-6$.

$$x^{2}-\frac{13}{6}x=-1$$

Divide $-\frac{13}{6}$, the coefficient of the $x$ term, by $2$ to get $-\frac{13}{12}$. Then add the square of $-\frac{13}{12}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=-1+\left(-\frac{13}{12}\right)^{2}$$

Square $-\frac{13}{12}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=-1+\frac{169}{144}$$

Add $-1$ to $\frac{169}{144}$.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{25}{144}$$

Factor $x^{2}-\frac{13}{6}x+\frac{169}{144}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{25}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{25}{144}}$$

Simplify.

$$x-\frac{13}{12}=\frac{5}{12}$$ $$x-\frac{13}{12}=-\frac{5}{12}$$

Add $\frac{13}{12}$ to both sides of the equation.

$$x=\frac{3}{2}$$ $$x=\frac{2}{3}$$