Solve 289x^2-578x-611=0

$$289 { x }^{ 2 } -578x-611 = 0$$

Solve for x

$$x=-\frac{13}{17}\approx -0.764705882$$

$$x = \frac{47}{17} = 2\frac{13}{17} \approx 2.764705882$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 289x^{2}+ax+bx-611. To find a and b, set up a system to be solved.

$$a+b=-578 ab=289\left(-611\right)=-176579$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -176579.

$$1,-176579 13,-13583 17,-10387 47,-3757 221,-799 289,-611$$

Calculate the sum for each pair.

$$1-176579=-176578 13-13583=-13570 17-10387=-10370 47-3757=-3710 221-799=-578 289-611=-322$$

The solution is the pair that gives sum -578.

$$a=-799 b=221$$

Rewrite 289x^{2}-578x-611 as \left(289x^{2}-799x\right)+\left(221x-611\right).

$$\left(289x^{2}-799x\right)+\left(221x-611\right)$$

Factor out 17x in the first and 13 in the second group.

$$17x\left(17x-47\right)+13\left(17x-47\right)$$

Factor out common term 17x-47 by using distributive property.

$$\left(17x-47\right)\left(17x+13\right)$$

To find equation solutions, solve 17x-47=0 and 17x+13=0.

$$x=\frac{47}{17} x=-\frac{13}{17}$$

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.

$$289x^{2}-578x-611=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 289 for a, -578 for b, and -611 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-578\right)±\sqrt{\left(-578\right)^{2}-4\times 289\left(-611\right)}}{2\times 289}$$

Square -578.

$$x=\frac{-\left(-578\right)±\sqrt{334084-4\times 289\left(-611\right)}}{2\times 289}$$

Multiply -4 times 289.

$$x=\frac{-\left(-578\right)±\sqrt{334084-1156\left(-611\right)}}{2\times 289}$$

Multiply -1156 times -611.

$$x=\frac{-\left(-578\right)±\sqrt{334084+706316}}{2\times 289}$$

Add 334084 to 706316.

$$x=\frac{-\left(-578\right)±\sqrt{1040400}}{2\times 289}$$

Take the square root of 1040400.

$$x=\frac{-\left(-578\right)±1020}{2\times 289}$$

The opposite of -578 is 578.

$$x=\frac{578±1020}{2\times 289}$$

Multiply 2 times 289.

$$x=\frac{578±1020}{578}$$

Now solve the equation x=\frac{578±1020}{578} when ± is plus. Add 578 to 1020.

$$x=\frac{1598}{578}$$

Reduce the fraction \frac{1598}{578} to lowest terms by extracting and canceling out 34.

$$x=\frac{47}{17}$$

Now solve the equation x=\frac{578±1020}{578} when ± is minus. Subtract 1020 from 578.

$$x=-\frac{442}{578}$$

Reduce the fraction \frac{-442}{578} to lowest terms by extracting and canceling out 34.

$$x=-\frac{13}{17}$$

The equation is now solved.

$$x=\frac{47}{17} x=-\frac{13}{17}$$

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.

$$289x^{2}-578x-611=0$$

Add 611 to both sides of the equation.

$$289x^{2}-578x-611-\left(-611\right)=-\left(-611\right)$$

Subtracting -611 from itself leaves 0.

$$289x^{2}-578x=-\left(-611\right)$$

Subtract -611 from 0.

$$289x^{2}-578x=611$$

Divide both sides by 289.

$$\frac{289x^{2}-578x}{289}=\frac{611}{289}$$

Dividing by 289 undoes the multiplication by 289.

$$x^{2}+\left(-\frac{578}{289}\right)x=\frac{611}{289}$$

Divide -578 by 289.

$$x^{2}-2x=\frac{611}{289}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{611}{289}+1$$

Add \frac{611}{289} to 1.

$$x^{2}-2x+1=\frac{900}{289}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{900}{289}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{900}{289}}$$

Simplify.

$$x-1=\frac{30}{17} x-1=-\frac{30}{17}$$

Add 1 to both sides of the equation.

$$x=\frac{47}{17} x=-\frac{13}{17}$$