Solve 97x^2-645x+800=0

$$97 { x }^{ 2 } -645x+800=0$$

Solve for x

$$x=5$$

$$x = \frac{160}{97} = 1\frac{63}{97} \approx 1.649484536$$

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.

$$97x^{2}-645x+800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 97 for a, -645 for b, and 800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-645\right)±\sqrt{\left(-645\right)^{2}-4\times 97\times 800}}{2\times 97}$$

Square -645.

$$x=\frac{-\left(-645\right)±\sqrt{416025-4\times 97\times 800}}{2\times 97}$$

Multiply -4 times 97.

$$x=\frac{-\left(-645\right)±\sqrt{416025-388\times 800}}{2\times 97}$$

Multiply -388 times 800.

$$x=\frac{-\left(-645\right)±\sqrt{416025-310400}}{2\times 97}$$

Add 416025 to -310400.

$$x=\frac{-\left(-645\right)±\sqrt{105625}}{2\times 97}$$

Take the square root of 105625.

$$x=\frac{-\left(-645\right)±325}{2\times 97}$$

The opposite of -645 is 645.

$$x=\frac{645±325}{2\times 97}$$

Multiply 2 times 97.

$$x=\frac{645±325}{194}$$

Now solve the equation x=\frac{645±325}{194} when ± is plus. Add 645 to 325.

$$x=\frac{970}{194}$$

Divide 970 by 194.

$$x=5$$

Now solve the equation x=\frac{645±325}{194} when ± is minus. Subtract 325 from 645.

$$x=\frac{320}{194}$$

Reduce the fraction \frac{320}{194} to lowest terms by extracting and canceling out 2.

$$x=\frac{160}{97}$$

The equation is now solved.

$$x=5 x=\frac{160}{97}$$

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.

$$97x^{2}-645x+800=0$$

Subtract 800 from both sides of the equation.

$$97x^{2}-645x+800-800=-800$$

Subtracting 800 from itself leaves 0.

$$97x^{2}-645x=-800$$

Divide both sides by 97.

$$\frac{97x^{2}-645x}{97}=-\frac{800}{97}$$

Dividing by 97 undoes the multiplication by 97.

$$x^{2}-\frac{645}{97}x=-\frac{800}{97}$$

Divide -\frac{645}{97}, the coefficient of the x term, by 2 to get -\frac{645}{194}. Then add the square of -\frac{645}{194} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{645}{97}x+\left(-\frac{645}{194}\right)^{2}=-\frac{800}{97}+\left(-\frac{645}{194}\right)^{2}$$

Square -\frac{645}{194} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{645}{97}x+\frac{416025}{37636}=-\frac{800}{97}+\frac{416025}{37636}$$

Add -\frac{800}{97} to \frac{416025}{37636} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{645}{97}x+\frac{416025}{37636}=\frac{105625}{37636}$$

Factor x^{2}-\frac{645}{97}x+\frac{416025}{37636}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{645}{194}\right)^{2}=\frac{105625}{37636}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{645}{194}\right)^{2}}=\sqrt{\frac{105625}{37636}}$$

Simplify.

$$x-\frac{645}{194}=\frac{325}{194} x-\frac{645}{194}=-\frac{325}{194}$$

Add \frac{645}{194} to both sides of the equation.

$$x=5 x=\frac{160}{97}$$