Solve 6(x^2+2)=7x

$$6 ( x ^ { 2 } + 2 ) = 7 x$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{239}i}{12}\approx 0.583333333+1.288302069i$$

$$x=\frac{-\sqrt{239}i+7}{12}\approx 0.583333333-1.288302069i$$

Steps Using the Quadratic Formula

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

$$6x^{2}+12=7x$$

Subtract 7x from both sides.

$$6x^{2}+12-7x=0$$

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.

$$6x^{2}-7x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -7 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\times 12}}{2\times 6}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6\times 12}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\times 12}}{2\times 6}$$

Multiply -24 times 12.

$$x=\frac{-\left(-7\right)±\sqrt{49-288}}{2\times 6}$$

Add 49 to -288.

$$x=\frac{-\left(-7\right)±\sqrt{-239}}{2\times 6}$$

Take the square root of -239.

$$x=\frac{-\left(-7\right)±\sqrt{239}i}{2\times 6}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{239}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{7±\sqrt{239}i}{12}$$

Now solve the equation x=\frac{7±\sqrt{239}i}{12} when ± is plus. Add 7 to i\sqrt{239}.

$$x=\frac{7+\sqrt{239}i}{12}$$

Now solve the equation x=\frac{7±\sqrt{239}i}{12} when ± is minus. Subtract i\sqrt{239} from 7.

$$x=\frac{-\sqrt{239}i+7}{12}$$

The equation is now solved.

$$x=\frac{7+\sqrt{239}i}{12} x=\frac{-\sqrt{239}i+7}{12}$$

Steps for Completing the Square

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

$$6x^{2}+12=7x$$

Subtract 7x from both sides.

$$6x^{2}+12-7x=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}-7x=-12$$

Divide both sides by 6.

$$\frac{6x^{2}-7x}{6}=-\frac{12}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{7}{6}x=-\frac{12}{6}$$

Divide -12 by 6.

$$x^{2}-\frac{7}{6}x=-2$$

Divide -\frac{7}{6}, the coefficient of the x term, by 2 to get -\frac{7}{12}. Then add the square of -\frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{6}x+\left(-\frac{7}{12}\right)^{2}=-2+\left(-\frac{7}{12}\right)^{2}$$

Square -\frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-2+\frac{49}{144}$$

Add -2 to \frac{49}{144}.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{239}{144}$$

Factor x^{2}-\frac{7}{6}x+\frac{49}{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}.

$$\left(x-\frac{7}{12}\right)^{2}=-\frac{239}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{12}\right)^{2}}=\sqrt{-\frac{239}{144}}$$

Simplify.

$$x-\frac{7}{12}=\frac{\sqrt{239}i}{12} x-\frac{7}{12}=-\frac{\sqrt{239}i}{12}$$

Add \frac{7}{12} to both sides of the equation.

$$x=\frac{7+\sqrt{239}i}{12} x=\frac{-\sqrt{239}i+7}{12}$$