Solve -3x^2+28x+39=0

$$-3 { x }^{ 2 } +28x+39=0$$

Solve for x

$$x = \frac{\sqrt{313} + 14}{3} \approx 10.563935338$$

$$x=\frac{14-\sqrt{313}}{3}\approx -1.230602004$$

Steps Using the Quadratic Formula

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.

$$-3x^{2}+28x+39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 28 for b, and 39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-3\right)\times 39}}{2\left(-3\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-3\right)\times 39}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-28±\sqrt{784+12\times 39}}{2\left(-3\right)}$$

Multiply 12 times 39.

$$x=\frac{-28±\sqrt{784+468}}{2\left(-3\right)}$$

Add 784 to 468.

$$x=\frac{-28±\sqrt{1252}}{2\left(-3\right)}$$

Take the square root of 1252.

$$x=\frac{-28±2\sqrt{313}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-28±2\sqrt{313}}{-6}$$

Now solve the equation x=\frac{-28±2\sqrt{313}}{-6} when ± is plus. Add -28 to 2\sqrt{313}.

$$x=\frac{2\sqrt{313}-28}{-6}$$

Divide -28+2\sqrt{313} by -6.

$$x=\frac{14-\sqrt{313}}{3}$$

Now solve the equation x=\frac{-28±2\sqrt{313}}{-6} when ± is minus. Subtract 2\sqrt{313} from -28.

$$x=\frac{-2\sqrt{313}-28}{-6}$$

Divide -28-2\sqrt{313} by -6.

$$x=\frac{\sqrt{313}+14}{3}$$

The equation is now solved.

$$x=\frac{14-\sqrt{313}}{3} x=\frac{\sqrt{313}+14}{3}$$

Steps for Completing the Square

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.

$$-3x^{2}+28x+39=0$$

Subtract 39 from both sides of the equation.

$$-3x^{2}+28x+39-39=-39$$

Subtracting 39 from itself leaves 0.

$$-3x^{2}+28x=-39$$

Divide both sides by -3.

$$\frac{-3x^{2}+28x}{-3}=-\frac{39}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{28}{-3}x=-\frac{39}{-3}$$

Divide 28 by -3.

$$x^{2}-\frac{28}{3}x=-\frac{39}{-3}$$

Divide -39 by -3.

$$x^{2}-\frac{28}{3}x=13$$

Divide -\frac{28}{3}, the coefficient of the x term, by 2 to get -\frac{14}{3}. Then add the square of -\frac{14}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{28}{3}x+\left(-\frac{14}{3}\right)^{2}=13+\left(-\frac{14}{3}\right)^{2}$$

Square -\frac{14}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{28}{3}x+\frac{196}{9}=13+\frac{196}{9}$$

Add 13 to \frac{196}{9}.

$$x^{2}-\frac{28}{3}x+\frac{196}{9}=\frac{313}{9}$$

Factor x^{2}-\frac{28}{3}x+\frac{196}{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}.

$$\left(x-\frac{14}{3}\right)^{2}=\frac{313}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{14}{3}\right)^{2}}=\sqrt{\frac{313}{9}}$$

Simplify.

$$x-\frac{14}{3}=\frac{\sqrt{313}}{3} x-\frac{14}{3}=-\frac{\sqrt{313}}{3}$$

Add \frac{14}{3} to both sides of the equation.

$$x=\frac{\sqrt{313}+14}{3} x=\frac{14-\sqrt{313}}{3}$$