Solve 4x^2+280x-2400=0

$$4 { x }^{ 2 } +280x-2400=0$$

Solve for x

$$x=5\sqrt{73}-35\approx 7.720018727$$

$$x=-5\sqrt{73}-35\approx -77.720018727$$

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.

$$4x^{2}+280x-2400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 280 for b, and -2400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-280±\sqrt{280^{2}-4\times 4\left(-2400\right)}}{2\times 4}$$

Square 280.

$$x=\frac{-280±\sqrt{78400-4\times 4\left(-2400\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-280±\sqrt{78400-16\left(-2400\right)}}{2\times 4}$$

Multiply -16 times -2400.

$$x=\frac{-280±\sqrt{78400+38400}}{2\times 4}$$

Add 78400 to 38400.

$$x=\frac{-280±\sqrt{116800}}{2\times 4}$$

Take the square root of 116800.

$$x=\frac{-280±40\sqrt{73}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-280±40\sqrt{73}}{8}$$

Now solve the equation x=\frac{-280±40\sqrt{73}}{8} when ± is plus. Add -280 to 40\sqrt{73}.

$$x=\frac{40\sqrt{73}-280}{8}$$

Divide -280+40\sqrt{73} by 8.

$$x=5\sqrt{73}-35$$

Now solve the equation x=\frac{-280±40\sqrt{73}}{8} when ± is minus. Subtract 40\sqrt{73} from -280.

$$x=\frac{-40\sqrt{73}-280}{8}$$

Divide -280-40\sqrt{73} by 8.

$$x=-5\sqrt{73}-35$$

The equation is now solved.

$$x=5\sqrt{73}-35 x=-5\sqrt{73}-35$$

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.

$$4x^{2}+280x-2400=0$$

Add 2400 to both sides of the equation.

$$4x^{2}+280x-2400-\left(-2400\right)=-\left(-2400\right)$$

Subtracting -2400 from itself leaves 0.

$$4x^{2}+280x=-\left(-2400\right)$$

Subtract -2400 from 0.

$$4x^{2}+280x=2400$$

Divide both sides by 4.

$$\frac{4x^{2}+280x}{4}=\frac{2400}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{280}{4}x=\frac{2400}{4}$$

Divide 280 by 4.

$$x^{2}+70x=\frac{2400}{4}$$

Divide 2400 by 4.

$$x^{2}+70x=600$$

Divide 70, the coefficient of the x term, by 2 to get 35. Then add the square of 35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+70x+35^{2}=600+35^{2}$$

Square 35.

$$x^{2}+70x+1225=600+1225$$

Add 600 to 1225.

$$x^{2}+70x+1225=1825$$

Factor x^{2}+70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+35\right)^{2}=1825$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+35\right)^{2}}=\sqrt{1825}$$

Simplify.

$$x+35=5\sqrt{73} x+35=-5\sqrt{73}$$

Subtract 35 from both sides of the equation.

$$x=5\sqrt{73}-35 x=-5\sqrt{73}-35$$