Solve 169x^2-1040x+351=0

$$169 { x }^{ 2 } -1040x+351=0$$

Solve for x

$$x = \frac{\sqrt{1249} + 40}{13} \approx 5.795476469$$

$$x=\frac{40-\sqrt{1249}}{13}\approx 0.358369685$$

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.

$$169x^{2}-1040x+351=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 169 for a, -1040 for b, and 351 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1040\right)±\sqrt{\left(-1040\right)^{2}-4\times 169\times 351}}{2\times 169}$$

Square -1040.

$$x=\frac{-\left(-1040\right)±\sqrt{1081600-4\times 169\times 351}}{2\times 169}$$

Multiply -4 times 169.

$$x=\frac{-\left(-1040\right)±\sqrt{1081600-676\times 351}}{2\times 169}$$

Multiply -676 times 351.

$$x=\frac{-\left(-1040\right)±\sqrt{1081600-237276}}{2\times 169}$$

Add 1081600 to -237276.

$$x=\frac{-\left(-1040\right)±\sqrt{844324}}{2\times 169}$$

Take the square root of 844324.

$$x=\frac{-\left(-1040\right)±26\sqrt{1249}}{2\times 169}$$

The opposite of -1040 is 1040.

$$x=\frac{1040±26\sqrt{1249}}{2\times 169}$$

Multiply 2 times 169.

$$x=\frac{1040±26\sqrt{1249}}{338}$$

Now solve the equation x=\frac{1040±26\sqrt{1249}}{338} when ± is plus. Add 1040 to 26\sqrt{1249}.

$$x=\frac{26\sqrt{1249}+1040}{338}$$

Divide 1040+26\sqrt{1249} by 338.

$$x=\frac{\sqrt{1249}+40}{13}$$

Now solve the equation x=\frac{1040±26\sqrt{1249}}{338} when ± is minus. Subtract 26\sqrt{1249} from 1040.

$$x=\frac{1040-26\sqrt{1249}}{338}$$

Divide 1040-26\sqrt{1249} by 338.

$$x=\frac{40-\sqrt{1249}}{13}$$

The equation is now solved.

$$x=\frac{\sqrt{1249}+40}{13} x=\frac{40-\sqrt{1249}}{13}$$

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.

$$169x^{2}-1040x+351=0$$

Subtract 351 from both sides of the equation.

$$169x^{2}-1040x+351-351=-351$$

Subtracting 351 from itself leaves 0.

$$169x^{2}-1040x=-351$$

Divide both sides by 169.

$$\frac{169x^{2}-1040x}{169}=-\frac{351}{169}$$

Dividing by 169 undoes the multiplication by 169.

$$x^{2}+\left(-\frac{1040}{169}\right)x=-\frac{351}{169}$$

Reduce the fraction \frac{-1040}{169} to lowest terms by extracting and canceling out 13.

$$x^{2}-\frac{80}{13}x=-\frac{351}{169}$$

Reduce the fraction \frac{-351}{169} to lowest terms by extracting and canceling out 13.

$$x^{2}-\frac{80}{13}x=-\frac{27}{13}$$

Divide -\frac{80}{13}, the coefficient of the x term, by 2 to get -\frac{40}{13}. Then add the square of -\frac{40}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{80}{13}x+\left(-\frac{40}{13}\right)^{2}=-\frac{27}{13}+\left(-\frac{40}{13}\right)^{2}$$

Square -\frac{40}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{80}{13}x+\frac{1600}{169}=-\frac{27}{13}+\frac{1600}{169}$$

Add -\frac{27}{13} to \frac{1600}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{80}{13}x+\frac{1600}{169}=\frac{1249}{169}$$

Factor x^{2}-\frac{80}{13}x+\frac{1600}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{40}{13}\right)^{2}=\frac{1249}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{40}{13}\right)^{2}}=\sqrt{\frac{1249}{169}}$$

Simplify.

$$x-\frac{40}{13}=\frac{\sqrt{1249}}{13} x-\frac{40}{13}=-\frac{\sqrt{1249}}{13}$$

Add \frac{40}{13} to both sides of the equation.

$$x=\frac{\sqrt{1249}+40}{13} x=\frac{40-\sqrt{1249}}{13}$$