Solve (5x - 16)/(7x - 18) = 2/3 तर ? | 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

$$\frac { 5 x - 16 } { 7 x - 18 } = \frac { 2 } { 3 } \quad x =$$

Solve for x (complex solution)

$x=\frac{-\sqrt{87}i+51}{28}\approx 1.821428571-0.33312068i$
$x=\frac{51+\sqrt{87}i}{28}\approx 1.821428571+0.33312068i$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to $\frac{18}{7}$ since division by zero is not defined. Multiply both sides of the equation by $3\left(7x-18\right)$, the least common multiple of $7x-18,3$.

$$3\left(5x-16\right)=\frac{2}{3}x\times 3\left(7x-18\right)$$

Use the distributive property to multiply $3$ by $5x-16$.

$$15x-48=\frac{2}{3}x\times 3\left(7x-18\right)$$

Multiply $\frac{2}{3}$ and $3$ to get $2$.

$$15x-48=2x\left(7x-18\right)$$

Use the distributive property to multiply $2x$ by $7x-18$.

$$15x-48=14x^{2}-36x$$

Subtract $14x^{2}$ from both sides.

$$15x-48-14x^{2}=-36x$$

Add $36x$ to both sides.

$$15x-48-14x^{2}+36x=0$$

Combine $15x$ and $36x$ to get $51x$.

$$51x-48-14x^{2}=0$$

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.

$$-14x^{2}+51x-48=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-14$ for $a$, $51$ for $b$, and $-48$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-51±\sqrt{51^{2}-4\left(-14\right)\left(-48\right)}}{2\left(-14\right)}$$

Square $51$.

$$x=\frac{-51±\sqrt{2601-4\left(-14\right)\left(-48\right)}}{2\left(-14\right)}$$

Multiply $-4$ times $-14$.

$$x=\frac{-51±\sqrt{2601+56\left(-48\right)}}{2\left(-14\right)}$$

Multiply $56$ times $-48$.

$$x=\frac{-51±\sqrt{2601-2688}}{2\left(-14\right)}$$

Add $2601$ to $-2688$.

$$x=\frac{-51±\sqrt{-87}}{2\left(-14\right)}$$

Take the square root of $-87$.

$$x=\frac{-51±\sqrt{87}i}{2\left(-14\right)}$$

Multiply $2$ times $-14$.

$$x=\frac{-51±\sqrt{87}i}{-28}$$

Now solve the equation $x=\frac{-51±\sqrt{87}i}{-28}$ when $±$ is plus. Add $-51$ to $i\sqrt{87}$.

$$x=\frac{-51+\sqrt{87}i}{-28}$$

Divide $-51+i\sqrt{87}$ by $-28$.

$$x=\frac{-\sqrt{87}i+51}{28}$$

Now solve the equation $x=\frac{-51±\sqrt{87}i}{-28}$ when $±$ is minus. Subtract $i\sqrt{87}$ from $-51$.

$$x=\frac{-\sqrt{87}i-51}{-28}$$

Divide $-51-i\sqrt{87}$ by $-28$.

$$x=\frac{51+\sqrt{87}i}{28}$$

The equation is now solved.

$$x=\frac{-\sqrt{87}i+51}{28}$$ $$x=\frac{51+\sqrt{87}i}{28}$$

Steps for Completing the Square

Variable $x$ cannot be equal to $\frac{18}{7}$ since division by zero is not defined. Multiply both sides of the equation by $3\left(7x-18\right)$, the least common multiple of $7x-18,3$.

$$3\left(5x-16\right)=\frac{2}{3}x\times 3\left(7x-18\right)$$

Use the distributive property to multiply $3$ by $5x-16$.

$$15x-48=\frac{2}{3}x\times 3\left(7x-18\right)$$

Multiply $\frac{2}{3}$ and $3$ to get $2$.

$$15x-48=2x\left(7x-18\right)$$

Use the distributive property to multiply $2x$ by $7x-18$.

$$15x-48=14x^{2}-36x$$

Subtract $14x^{2}$ from both sides.

$$15x-48-14x^{2}=-36x$$

Add $36x$ to both sides.

$$15x-48-14x^{2}+36x=0$$

Combine $15x$ and $36x$ to get $51x$.

$$51x-48-14x^{2}=0$$

Add $48$ to both sides. Anything plus zero gives itself.

$$51x-14x^{2}=48$$

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 $x^{2}+bx=c$.

$$-14x^{2}+51x=48$$

Divide both sides by $-14$.

$$\frac{-14x^{2}+51x}{-14}=\frac{48}{-14}$$

Dividing by $-14$ undoes the multiplication by $-14$.

$$x^{2}+\frac{51}{-14}x=\frac{48}{-14}$$

Divide $51$ by $-14$.

$$x^{2}-\frac{51}{14}x=\frac{48}{-14}$$

Reduce the fraction $\frac{48}{-14}$ to lowest terms by extracting and canceling out $2$.

$$x^{2}-\frac{51}{14}x=-\frac{24}{7}$$

Divide $-\frac{51}{14}$, the coefficient of the $x$ term, by $2$ to get $-\frac{51}{28}$. Then add the square of $-\frac{51}{28}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{51}{14}x+\left(-\frac{51}{28}\right)^{2}=-\frac{24}{7}+\left(-\frac{51}{28}\right)^{2}$$

Square $-\frac{51}{28}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{51}{14}x+\frac{2601}{784}=-\frac{24}{7}+\frac{2601}{784}$$

Add $-\frac{24}{7}$ to $\frac{2601}{784}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{51}{14}x+\frac{2601}{784}=-\frac{87}{784}$$

Factor $x^{2}-\frac{51}{14}x+\frac{2601}{784}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{51}{28}\right)^{2}=-\frac{87}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{51}{28}\right)^{2}}=\sqrt{-\frac{87}{784}}$$

Simplify.

$$x-\frac{51}{28}=\frac{\sqrt{87}i}{28}$$ $$x-\frac{51}{28}=-\frac{\sqrt{87}i}{28}$$

Add $\frac{51}{28}$ to both sides of the equation.

$$x=\frac{51+\sqrt{87}i}{28}$$ $$x=\frac{-\sqrt{87}i+51}{28}$$