Solve 3(5x - 2) = 11x Find | 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

$$3 ( 5 x - 2 ) = 11 x x$$

Solve for x (complex solution)

$x=\frac{-\sqrt{39}i+15}{22}\approx 0.681818182-0.283863545i$
$x=\frac{15+\sqrt{39}i}{22}\approx 0.681818182+0.283863545i$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

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

$$3\left(5x-2\right)=11x^{2}$$

Use the distributive property to multiply $3$ by $5x-2$.

$$15x-6=11x^{2}$$

Subtract $11x^{2}$ from both sides.

$$15x-6-11x^{2}=0$$

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.

$$-11x^{2}+15x-6=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-11$ for $a$, $15$ for $b$, and $-6$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-11\right)\left(-6\right)}}{2\left(-11\right)}$$

Square $15$.

$$x=\frac{-15±\sqrt{225-4\left(-11\right)\left(-6\right)}}{2\left(-11\right)}$$

Multiply $-4$ times $-11$.

$$x=\frac{-15±\sqrt{225+44\left(-6\right)}}{2\left(-11\right)}$$

Multiply $44$ times $-6$.

$$x=\frac{-15±\sqrt{225-264}}{2\left(-11\right)}$$

Add $225$ to $-264$.

$$x=\frac{-15±\sqrt{-39}}{2\left(-11\right)}$$

Take the square root of $-39$.

$$x=\frac{-15±\sqrt{39}i}{2\left(-11\right)}$$

Multiply $2$ times $-11$.

$$x=\frac{-15±\sqrt{39}i}{-22}$$

Now solve the equation $x=\frac{-15±\sqrt{39}i}{-22}$ when $±$ is plus. Add $-15$ to $i\sqrt{39}$.

$$x=\frac{-15+\sqrt{39}i}{-22}$$

Divide $-15+i\sqrt{39}$ by $-22$.

$$x=\frac{-\sqrt{39}i+15}{22}$$

Now solve the equation $x=\frac{-15±\sqrt{39}i}{-22}$ when $±$ is minus. Subtract $i\sqrt{39}$ from $-15$.

$$x=\frac{-\sqrt{39}i-15}{-22}$$

Divide $-15-i\sqrt{39}$ by $-22$.

$$x=\frac{15+\sqrt{39}i}{22}$$

The equation is now solved.

$$x=\frac{-\sqrt{39}i+15}{22}$$ $$x=\frac{15+\sqrt{39}i}{22}$$

Steps for Completing the Square

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

$$3\left(5x-2\right)=11x^{2}$$

Use the distributive property to multiply $3$ by $5x-2$.

$$15x-6=11x^{2}$$

Subtract $11x^{2}$ from both sides.

$$15x-6-11x^{2}=0$$

Add $6$ to both sides. Anything plus zero gives itself.

$$15x-11x^{2}=6$$

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

$$-11x^{2}+15x=6$$

Divide both sides by $-11$.

$$\frac{-11x^{2}+15x}{-11}=\frac{6}{-11}$$

Dividing by $-11$ undoes the multiplication by $-11$.

$$x^{2}+\frac{15}{-11}x=\frac{6}{-11}$$

Divide $15$ by $-11$.

$$x^{2}-\frac{15}{11}x=\frac{6}{-11}$$

Divide $6$ by $-11$.

$$x^{2}-\frac{15}{11}x=-\frac{6}{11}$$

Divide $-\frac{15}{11}$, the coefficient of the $x$ term, by $2$ to get $-\frac{15}{22}$. Then add the square of $-\frac{15}{22}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{11}x+\left(-\frac{15}{22}\right)^{2}=-\frac{6}{11}+\left(-\frac{15}{22}\right)^{2}$$

Square $-\frac{15}{22}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{11}x+\frac{225}{484}=-\frac{6}{11}+\frac{225}{484}$$

Add $-\frac{6}{11}$ to $\frac{225}{484}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{11}x+\frac{225}{484}=-\frac{39}{484}$$

Factor $x^{2}-\frac{15}{11}x+\frac{225}{484}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{15}{22}\right)^{2}=-\frac{39}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{22}\right)^{2}}=\sqrt{-\frac{39}{484}}$$

Simplify.

$$x-\frac{15}{22}=\frac{\sqrt{39}i}{22}$$ $$x-\frac{15}{22}=-\frac{\sqrt{39}i}{22}$$

Add $\frac{15}{22}$ to both sides of the equation.

$$x=\frac{15+\sqrt{39}i}{22}$$ $$x=\frac{-\sqrt{39}i+15}{22}$$