How to solve 10x - 17 = - 3/x

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

Solve 10x - 17 = - 3/x | Equatix

Question

$$10x-17=\frac{-3}{x}$$

Solve for x

$x=\frac{1}{5}=0.2$
$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $x$.

$$10xx+x\left(-17\right)=-3$$

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

$$10x^{2}+x\left(-17\right)=-3$$

Add $3$ to both sides.

$$10x^{2}+x\left(-17\right)+3=0$$

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.

$$10x^{2}-17x+3=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $10$ for $a$, $-17$ for $b$, and $3$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 10\times 3}}{2\times 10}$$

Square $-17$.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 10\times 3}}{2\times 10}$$

Multiply $-4$ times $10$.

$$x=\frac{-\left(-17\right)±\sqrt{289-40\times 3}}{2\times 10}$$

Multiply $-40$ times $3$.

$$x=\frac{-\left(-17\right)±\sqrt{289-120}}{2\times 10}$$

Add $289$ to $-120$.

$$x=\frac{-\left(-17\right)±\sqrt{169}}{2\times 10}$$

Take the square root of $169$.

$$x=\frac{-\left(-17\right)±13}{2\times 10}$$

The opposite of $-17$ is $17$.

$$x=\frac{17±13}{2\times 10}$$

Multiply $2$ times $10$.

$$x=\frac{17±13}{20}$$

Now solve the equation $x=\frac{17±13}{20}$ when $±$ is plus. Add $17$ to $13$.

$$x=\frac{30}{20}$$

Reduce the fraction $\frac{30}{20}$ to lowest terms by extracting and canceling out $10$.

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

Now solve the equation $x=\frac{17±13}{20}$ when $±$ is minus. Subtract $13$ from $17$.

$$x=\frac{4}{20}$$

Reduce the fraction $\frac{4}{20}$ to lowest terms by extracting and canceling out $4$.

$$x=\frac{1}{5}$$

The equation is now solved.

$$x=\frac{3}{2}$$ $$x=\frac{1}{5}$$

Steps for Completing the Square

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $x$.

$$10xx+x\left(-17\right)=-3$$

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

$$10x^{2}+x\left(-17\right)=-3$$

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

$$10x^{2}-17x=-3$$

Divide both sides by $10$.

$$\frac{10x^{2}-17x}{10}=-\frac{3}{10}$$

Dividing by $10$ undoes the multiplication by $10$.

$$x^{2}-\frac{17}{10}x=-\frac{3}{10}$$

Divide $-\frac{17}{10}$, the coefficient of the $x$ term, by $2$ to get $-\frac{17}{20}$. Then add the square of $-\frac{17}{20}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{10}x+\left(-\frac{17}{20}\right)^{2}=-\frac{3}{10}+\left(-\frac{17}{20}\right)^{2}$$

Square $-\frac{17}{20}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{10}x+\frac{289}{400}=-\frac{3}{10}+\frac{289}{400}$$

Add $-\frac{3}{10}$ to $\frac{289}{400}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{17}{10}x+\frac{289}{400}=\frac{169}{400}$$

Factor $x^{2}-\frac{17}{10}x+\frac{289}{400}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{17}{20}\right)^{2}=\frac{169}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{20}\right)^{2}}=\sqrt{\frac{169}{400}}$$

Simplify.

$$x-\frac{17}{20}=\frac{13}{20}$$ $$x-\frac{17}{20}=-\frac{13}{20}$$

Add $\frac{17}{20}$ to both sides of the equation.

$$x=\frac{3}{2}$$ $$x=\frac{1}{5}$$

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Solve for x

Steps Using the Quadratic Formula

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Steps Using the Quadratic Formula

Steps for Completing the Square