Solve 3x^2+17x-2400=0

$$3 { x }^{ 2 } +17x-2400=0$$

Solve for x

$$x = \frac{\sqrt{29089} - 17}{6} \approx 25.592496079$$

$$x=\frac{-\sqrt{29089}-17}{6}\approx -31.259162746$$

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.

$$3x^{2}+17x-2400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 17 for b, and -2400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 3\left(-2400\right)}}{2\times 3}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 3\left(-2400\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-17±\sqrt{289-12\left(-2400\right)}}{2\times 3}$$

Multiply -12 times -2400.

$$x=\frac{-17±\sqrt{289+28800}}{2\times 3}$$

Add 289 to 28800.

$$x=\frac{-17±\sqrt{29089}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-17±\sqrt{29089}}{6}$$

Now solve the equation x=\frac{-17±\sqrt{29089}}{6} when ± is plus. Add -17 to \sqrt{29089}.

$$x=\frac{\sqrt{29089}-17}{6}$$

Now solve the equation x=\frac{-17±\sqrt{29089}}{6} when ± is minus. Subtract \sqrt{29089} from -17.

$$x=\frac{-\sqrt{29089}-17}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{29089}-17}{6} x=\frac{-\sqrt{29089}-17}{6}$$

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.

$$3x^{2}+17x-2400=0$$

Add 2400 to both sides of the equation.

$$3x^{2}+17x-2400-\left(-2400\right)=-\left(-2400\right)$$

Subtracting -2400 from itself leaves 0.

$$3x^{2}+17x=-\left(-2400\right)$$

Subtract -2400 from 0.

$$3x^{2}+17x=2400$$

Divide both sides by 3.

$$\frac{3x^{2}+17x}{3}=\frac{2400}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{17}{3}x=\frac{2400}{3}$$

Divide 2400 by 3.

$$x^{2}+\frac{17}{3}x=800$$

Divide \frac{17}{3}, the coefficient of the x term, by 2 to get \frac{17}{6}. Then add the square of \frac{17}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{17}{3}x+\left(\frac{17}{6}\right)^{2}=800+\left(\frac{17}{6}\right)^{2}$$

Square \frac{17}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{17}{3}x+\frac{289}{36}=800+\frac{289}{36}$$

Add 800 to \frac{289}{36}.

$$x^{2}+\frac{17}{3}x+\frac{289}{36}=\frac{29089}{36}$$

Factor x^{2}+\frac{17}{3}x+\frac{289}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{17}{6}\right)^{2}=\frac{29089}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{6}\right)^{2}}=\sqrt{\frac{29089}{36}}$$

Simplify.

$$x+\frac{17}{6}=\frac{\sqrt{29089}}{6} x+\frac{17}{6}=-\frac{\sqrt{29089}}{6}$$

Subtract \frac{17}{6} from both sides of the equation.

$$x=\frac{\sqrt{29089}-17}{6} x=\frac{-\sqrt{29089}-17}{6}$$