Solve 75x^2-766x+80=0

$$75 { x }^{ 2 } -766x+80=0$$

Solve for x

$$x = \frac{\sqrt{140689} + 383}{75} \approx 10.107804315$$

$$x=\frac{383-\sqrt{140689}}{75}\approx 0.105529018$$

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.

$$75x^{2}-766x+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 75 for a, -766 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-766\right)±\sqrt{\left(-766\right)^{2}-4\times 75\times 80}}{2\times 75}$$

Square -766.

$$x=\frac{-\left(-766\right)±\sqrt{586756-4\times 75\times 80}}{2\times 75}$$

Multiply -4 times 75.

$$x=\frac{-\left(-766\right)±\sqrt{586756-300\times 80}}{2\times 75}$$

Multiply -300 times 80.

$$x=\frac{-\left(-766\right)±\sqrt{586756-24000}}{2\times 75}$$

Add 586756 to -24000.

$$x=\frac{-\left(-766\right)±\sqrt{562756}}{2\times 75}$$

Take the square root of 562756.

$$x=\frac{-\left(-766\right)±2\sqrt{140689}}{2\times 75}$$

The opposite of -766 is 766.

$$x=\frac{766±2\sqrt{140689}}{2\times 75}$$

Multiply 2 times 75.

$$x=\frac{766±2\sqrt{140689}}{150}$$

Now solve the equation x=\frac{766±2\sqrt{140689}}{150} when ± is plus. Add 766 to 2\sqrt{140689}.

$$x=\frac{2\sqrt{140689}+766}{150}$$

Divide 766+2\sqrt{140689} by 150.

$$x=\frac{\sqrt{140689}+383}{75}$$

Now solve the equation x=\frac{766±2\sqrt{140689}}{150} when ± is minus. Subtract 2\sqrt{140689} from 766.

$$x=\frac{766-2\sqrt{140689}}{150}$$

Divide 766-2\sqrt{140689} by 150.

$$x=\frac{383-\sqrt{140689}}{75}$$

The equation is now solved.

$$x=\frac{\sqrt{140689}+383}{75} x=\frac{383-\sqrt{140689}}{75}$$

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.

$$75x^{2}-766x+80=0$$

Subtract 80 from both sides of the equation.

$$75x^{2}-766x+80-80=-80$$

Subtracting 80 from itself leaves 0.

$$75x^{2}-766x=-80$$

Divide both sides by 75.

$$\frac{75x^{2}-766x}{75}=-\frac{80}{75}$$

Dividing by 75 undoes the multiplication by 75.

$$x^{2}-\frac{766}{75}x=-\frac{80}{75}$$

Reduce the fraction \frac{-80}{75} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{766}{75}x=-\frac{16}{15}$$

Divide -\frac{766}{75}, the coefficient of the x term, by 2 to get -\frac{383}{75}. Then add the square of -\frac{383}{75} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{766}{75}x+\left(-\frac{383}{75}\right)^{2}=-\frac{16}{15}+\left(-\frac{383}{75}\right)^{2}$$

Square -\frac{383}{75} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{766}{75}x+\frac{146689}{5625}=-\frac{16}{15}+\frac{146689}{5625}$$

Add -\frac{16}{15} to \frac{146689}{5625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{766}{75}x+\frac{146689}{5625}=\frac{140689}{5625}$$

Factor x^{2}-\frac{766}{75}x+\frac{146689}{5625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{383}{75}\right)^{2}=\frac{140689}{5625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{383}{75}\right)^{2}}=\sqrt{\frac{140689}{5625}}$$

Simplify.

$$x-\frac{383}{75}=\frac{\sqrt{140689}}{75} x-\frac{383}{75}=-\frac{\sqrt{140689}}{75}$$

Add \frac{383}{75} to both sides of the equation.

$$x=\frac{\sqrt{140689}+383}{75} x=\frac{383-\sqrt{140689}}{75}$$