Solve 6794+x^2=165x

$$6794+ { x }^{ 2 } =165x$$

Solve for x

$$x=79$$

$$x=86$$

Steps Using the Quadratic Formula

Subtract 165x from both sides.

$$6794+x^{2}-165x=0$$

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.

$$x^{2}-165x+6794=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -165 for b, and 6794 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-165\right)±\sqrt{\left(-165\right)^{2}-4\times 6794}}{2}$$

Square -165.

$$x=\frac{-\left(-165\right)±\sqrt{27225-4\times 6794}}{2}$$

Multiply -4 times 6794.

$$x=\frac{-\left(-165\right)±\sqrt{27225-27176}}{2}$$

Add 27225 to -27176.

$$x=\frac{-\left(-165\right)±\sqrt{49}}{2}$$

Take the square root of 49.

$$x=\frac{-\left(-165\right)±7}{2}$$

The opposite of -165 is 165.

$$x=\frac{165±7}{2}$$

Now solve the equation x=\frac{165±7}{2} when ± is plus. Add 165 to 7.

$$x=\frac{172}{2}$$

Divide 172 by 2.

$$x=86$$

Now solve the equation x=\frac{165±7}{2} when ± is minus. Subtract 7 from 165.

$$x=\frac{158}{2}$$

Divide 158 by 2.

$$x=79$$

The equation is now solved.

$$x=86 x=79$$

Steps for Completing the Square

Subtract 165x from both sides.

$$6794+x^{2}-165x=0$$

Subtract 6794 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-165x=-6794$$

Divide -165, the coefficient of the x term, by 2 to get -\frac{165}{2}. Then add the square of -\frac{165}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-165x+\left(-\frac{165}{2}\right)^{2}=-6794+\left(-\frac{165}{2}\right)^{2}$$

Square -\frac{165}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-165x+\frac{27225}{4}=-6794+\frac{27225}{4}$$

Add -6794 to \frac{27225}{4}.

$$x^{2}-165x+\frac{27225}{4}=\frac{49}{4}$$

Factor x^{2}-165x+\frac{27225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{165}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{165}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{165}{2}=\frac{7}{2} x-\frac{165}{2}=-\frac{7}{2}$$

Add \frac{165}{2} to both sides of the equation.

$$x=86 x=79$$