Solve 780x^2-28600x-38200=0

$$780 { x }^{ 2 } -28600x-38200=0$$

Solve for x

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3}\approx 37.956928062$$

$$x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}\approx -1.290261396$$

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.

$$780x^{2}-28600x-38200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 780 for a, -28600 for b, and -38200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28600\right)±\sqrt{\left(-28600\right)^{2}-4\times 780\left(-38200\right)}}{2\times 780}$$

Square -28600.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-4\times 780\left(-38200\right)}}{2\times 780}$$

Multiply -4 times 780.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-3120\left(-38200\right)}}{2\times 780}$$

Multiply -3120 times -38200.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000+119184000}}{2\times 780}$$

Add 817960000 to 119184000.

$$x=\frac{-\left(-28600\right)±\sqrt{937144000}}{2\times 780}$$

Take the square root of 937144000.

$$x=\frac{-\left(-28600\right)±40\sqrt{585715}}{2\times 780}$$

The opposite of -28600 is 28600.

$$x=\frac{28600±40\sqrt{585715}}{2\times 780}$$

Multiply 2 times 780.

$$x=\frac{28600±40\sqrt{585715}}{1560}$$

Now solve the equation x=\frac{28600±40\sqrt{585715}}{1560} when ± is plus. Add 28600 to 40\sqrt{585715}.

$$x=\frac{40\sqrt{585715}+28600}{1560}$$

Divide 28600+40\sqrt{585715} by 1560.

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3}$$

Now solve the equation x=\frac{28600±40\sqrt{585715}}{1560} when ± is minus. Subtract 40\sqrt{585715} from 28600.

$$x=\frac{28600-40\sqrt{585715}}{1560}$$

Divide 28600-40\sqrt{585715} by 1560.

$$x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3} x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}$$

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.

$$780x^{2}-28600x-38200=0$$

Add 38200 to both sides of the equation.

$$780x^{2}-28600x-38200-\left(-38200\right)=-\left(-38200\right)$$

Subtracting -38200 from itself leaves 0.

$$780x^{2}-28600x=-\left(-38200\right)$$

Subtract -38200 from 0.

$$780x^{2}-28600x=38200$$

Divide both sides by 780.

$$\frac{780x^{2}-28600x}{780}=\frac{38200}{780}$$

Dividing by 780 undoes the multiplication by 780.

$$x^{2}+\left(-\frac{28600}{780}\right)x=\frac{38200}{780}$$

Reduce the fraction \frac{-28600}{780} to lowest terms by extracting and canceling out 260.

$$x^{2}-\frac{110}{3}x=\frac{38200}{780}$$

Reduce the fraction \frac{38200}{780} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{110}{3}x=\frac{1910}{39}$$

Divide -\frac{110}{3}, the coefficient of the x term, by 2 to get -\frac{55}{3}. Then add the square of -\frac{55}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{110}{3}x+\left(-\frac{55}{3}\right)^{2}=\frac{1910}{39}+\left(-\frac{55}{3}\right)^{2}$$

Square -\frac{55}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{1910}{39}+\frac{3025}{9}$$

Add \frac{1910}{39} to \frac{3025}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{45055}{117}$$

Factor x^{2}-\frac{110}{3}x+\frac{3025}{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}.

$$\left(x-\frac{55}{3}\right)^{2}=\frac{45055}{117}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{3}\right)^{2}}=\sqrt{\frac{45055}{117}}$$

Simplify.

$$x-\frac{55}{3}=\frac{\sqrt{585715}}{39} x-\frac{55}{3}=-\frac{\sqrt{585715}}{39}$$

Add \frac{55}{3} to both sides of the equation.

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3} x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}$$