Solve 5x^2-113x+430=0

$$5 { x }^{ 2 } -113x+430=0$$

Solve for x

$$x = \frac{\sqrt{4169} + 113}{10} \approx 17.756779383$$

$$x = \frac{113 - \sqrt{4169}}{10} \approx 4.843220617$$

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.

$$5x^{2}-113x+430=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -113 for b, and 430 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-113\right)±\sqrt{\left(-113\right)^{2}-4\times 5\times 430}}{2\times 5}$$

Square -113.

$$x=\frac{-\left(-113\right)±\sqrt{12769-4\times 5\times 430}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-113\right)±\sqrt{12769-20\times 430}}{2\times 5}$$

Multiply -20 times 430.

$$x=\frac{-\left(-113\right)±\sqrt{12769-8600}}{2\times 5}$$

Add 12769 to -8600.

$$x=\frac{-\left(-113\right)±\sqrt{4169}}{2\times 5}$$

The opposite of -113 is 113.

$$x=\frac{113±\sqrt{4169}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{113±\sqrt{4169}}{10}$$

Now solve the equation x=\frac{113±\sqrt{4169}}{10} when ± is plus. Add 113 to \sqrt{4169}.

$$x=\frac{\sqrt{4169}+113}{10}$$

Now solve the equation x=\frac{113±\sqrt{4169}}{10} when ± is minus. Subtract \sqrt{4169} from 113.

$$x=\frac{113-\sqrt{4169}}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{4169}+113}{10} x=\frac{113-\sqrt{4169}}{10}$$

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.

$$5x^{2}-113x+430=0$$

Subtract 430 from both sides of the equation.

$$5x^{2}-113x+430-430=-430$$

Subtracting 430 from itself leaves 0.

$$5x^{2}-113x=-430$$

Divide both sides by 5.

$$\frac{5x^{2}-113x}{5}=-\frac{430}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{113}{5}x=-\frac{430}{5}$$

Divide -430 by 5.

$$x^{2}-\frac{113}{5}x=-86$$

Divide -\frac{113}{5}, the coefficient of the x term, by 2 to get -\frac{113}{10}. Then add the square of -\frac{113}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{113}{5}x+\left(-\frac{113}{10}\right)^{2}=-86+\left(-\frac{113}{10}\right)^{2}$$

Square -\frac{113}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{113}{5}x+\frac{12769}{100}=-86+\frac{12769}{100}$$

Add -86 to \frac{12769}{100}.

$$x^{2}-\frac{113}{5}x+\frac{12769}{100}=\frac{4169}{100}$$

Factor x^{2}-\frac{113}{5}x+\frac{12769}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{113}{10}\right)^{2}=\frac{4169}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{113}{10}\right)^{2}}=\sqrt{\frac{4169}{100}}$$

Simplify.

$$x-\frac{113}{10}=\frac{\sqrt{4169}}{10} x-\frac{113}{10}=-\frac{\sqrt{4169}}{10}$$

Add \frac{113}{10} to both sides of the equation.

$$x=\frac{\sqrt{4169}+113}{10} x=\frac{113-\sqrt{4169}}{10}$$