Solve (2x ^ 2 + x)/3 - (2 - 3x)/4 = (x ^ 2 - 6)/6 | 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

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

Solve for x

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

Steps Using Factoring By Grouping

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using Factoring By Grouping

Multiply both sides of the equation by $12$, the least common multiple of $3,4,6$.

$$4\left(2x^{2}+x\right)-3\left(2-3x\right)=2\left(x^{2}-6\right)$$

Use the distributive property to multiply $4$ by $2x^{2}+x$.

$$8x^{2}+4x-3\left(2-3x\right)=2\left(x^{2}-6\right)$$

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

$$8x^{2}+4x-6+9x=2\left(x^{2}-6\right)$$

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

$$8x^{2}+13x-6=2\left(x^{2}-6\right)$$

Use the distributive property to multiply $2$ by $x^{2}-6$.

$$8x^{2}+13x-6=2x^{2}-12$$

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

$$8x^{2}+13x-6-2x^{2}=-12$$

Combine $8x^{2}$ and $-2x^{2}$ to get $6x^{2}$.

$$6x^{2}+13x-6=-12$$

Add $12$ to both sides.

$$6x^{2}+13x-6+12=0$$

Add $-6$ and $12$ to get $6$.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as $6x^{2}+ax+bx+6$. To find $a$ and $b$, set up a system to be solved.

$$a+b=13$$ $$ab=6\times 6=36$$

Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. List all such integer pairs that give product $36$.

$$1,36$$ $$2,18$$ $$3,12$$ $$4,9$$ $$6,6$$

Calculate the sum for each pair.

$$1+36=37$$ $$2+18=20$$ $$3+12=15$$ $$4+9=13$$ $$6+6=12$$

The solution is the pair that gives sum $13$.

$$a=4$$ $$b=9$$

Rewrite $6x^{2}+13x+6$ as $\left(6x^{2}+4x\right)+\left(9x+6\right)$.

$$\left(6x^{2}+4x\right)+\left(9x+6\right)$$

Factor out $2x$ in the first and $3$ in the second group.

$$2x\left(3x+2\right)+3\left(3x+2\right)$$

Factor out common term $3x+2$ by using distributive property.

$$\left(3x+2\right)\left(2x+3\right)$$

To find equation solutions, solve $3x+2=0$ and $2x+3=0$.

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

Steps Using the Quadratic Formula

Multiply both sides of the equation by $12$, the least common multiple of $3,4,6$.

$$4\left(2x^{2}+x\right)-3\left(2-3x\right)=2\left(x^{2}-6\right)$$

Use the distributive property to multiply $4$ by $2x^{2}+x$.

$$8x^{2}+4x-3\left(2-3x\right)=2\left(x^{2}-6\right)$$

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

$$8x^{2}+4x-6+9x=2\left(x^{2}-6\right)$$

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

$$8x^{2}+13x-6=2\left(x^{2}-6\right)$$

Use the distributive property to multiply $2$ by $x^{2}-6$.

$$8x^{2}+13x-6=2x^{2}-12$$

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

$$8x^{2}+13x-6-2x^{2}=-12$$

Combine $8x^{2}$ and $-2x^{2}$ to get $6x^{2}$.

$$6x^{2}+13x-6=-12$$

Add $12$ to both sides.

$$6x^{2}+13x-6+12=0$$

Add $-6$ and $12$ to get $6$.

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

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, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-13±\sqrt{13^{2}-4\times 6\times 6}}{2\times 6}$$

Square $13$.

$$x=\frac{-13±\sqrt{169-4\times 6\times 6}}{2\times 6}$$

Multiply $-4$ times $6$.

$$x=\frac{-13±\sqrt{169-24\times 6}}{2\times 6}$$

Multiply $-24$ times $6$.

$$x=\frac{-13±\sqrt{169-144}}{2\times 6}$$

Add $169$ to $-144$.

$$x=\frac{-13±\sqrt{25}}{2\times 6}$$

Take the square root of $25$.

$$x=\frac{-13±5}{2\times 6}$$

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

Multiply both sides of the equation by $12$, the least common multiple of $3,4,6$.

$$4\left(2x^{2}+x\right)-3\left(2-3x\right)=2\left(x^{2}-6\right)$$

Use the distributive property to multiply $4$ by $2x^{2}+x$.

$$8x^{2}+4x-3\left(2-3x\right)=2\left(x^{2}-6\right)$$

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

$$8x^{2}+4x-6+9x=2\left(x^{2}-6\right)$$

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

$$8x^{2}+13x-6=2\left(x^{2}-6\right)$$

Use the distributive property to multiply $2$ by $x^{2}-6$.

$$8x^{2}+13x-6=2x^{2}-12$$

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

$$8x^{2}+13x-6-2x^{2}=-12$$

Combine $8x^{2}$ and $-2x^{2}$ to get $6x^{2}$.

$$6x^{2}+13x-6=-12$$

Add $6$ to both sides.

$$6x^{2}+13x=-12+6$$

Add $-12$ and $6$ to get $-6$.

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

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 $-6$ 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.

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

Subtract $\frac{13}{12}$ from both sides of the equation.

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