Solve 28x=6x^2+80

$$28 x = 6 x ^ { 2 } + 80$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{71}i}{3}\approx 2.333333333+2.808716591i$$

$$x=\frac{-\sqrt{71}i+7}{3}\approx 2.333333333-2.808716591i$$

Steps Using the Quadratic Formula

Subtract 6x^{2} from both sides.

$$28x-6x^{2}=80$$

Subtract 80 from both sides.

$$28x-6x^{2}-80=0$$

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.

$$-6x^{2}+28x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 28 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-6\right)\left(-80\right)}}{2\left(-6\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-6\right)\left(-80\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-28±\sqrt{784+24\left(-80\right)}}{2\left(-6\right)}$$

Multiply 24 times -80.

$$x=\frac{-28±\sqrt{784-1920}}{2\left(-6\right)}$$

Add 784 to -1920.

$$x=\frac{-28±\sqrt{-1136}}{2\left(-6\right)}$$

Take the square root of -1136.

$$x=\frac{-28±4\sqrt{71}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-28±4\sqrt{71}i}{-12}$$

Now solve the equation x=\frac{-28±4\sqrt{71}i}{-12} when ± is plus. Add -28 to 4i\sqrt{71}.

$$x=\frac{-28+4\sqrt{71}i}{-12}$$

Divide -28+4i\sqrt{71} by -12.

$$x=\frac{-\sqrt{71}i+7}{3}$$

Now solve the equation x=\frac{-28±4\sqrt{71}i}{-12} when ± is minus. Subtract 4i\sqrt{71} from -28.

$$x=\frac{-4\sqrt{71}i-28}{-12}$$

Divide -28-4i\sqrt{71} by -12.

$$x=\frac{7+\sqrt{71}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{71}i+7}{3} x=\frac{7+\sqrt{71}i}{3}$$

Steps for Completing the Square

Subtract 6x^{2} from both sides.

$$28x-6x^{2}=80$$

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.

$$-6x^{2}+28x=80$$

Divide both sides by -6.

$$\frac{-6x^{2}+28x}{-6}=\frac{80}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{28}{-6}x=\frac{80}{-6}$$

Reduce the fraction \frac{28}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{14}{3}x=\frac{80}{-6}$$

Reduce the fraction \frac{80}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{14}{3}x=-\frac{40}{3}$$

Divide -\frac{14}{3}, the coefficient of the x term, by 2 to get -\frac{7}{3}. Then add the square of -\frac{7}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=-\frac{40}{3}+\left(-\frac{7}{3}\right)^{2}$$

Square -\frac{7}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-\frac{40}{3}+\frac{49}{9}$$

Add -\frac{40}{3} to \frac{49}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-\frac{71}{9}$$

Factor x^{2}-\frac{14}{3}x+\frac{49}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{3}\right)^{2}=-\frac{71}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{-\frac{71}{9}}$$

Simplify.

$$x-\frac{7}{3}=\frac{\sqrt{71}i}{3} x-\frac{7}{3}=-\frac{\sqrt{71}i}{3}$$

Add \frac{7}{3} to both sides of the equation.

$$x=\frac{7+\sqrt{71}i}{3} x=\frac{-\sqrt{71}i+7}{3}$$