Solve x^2-3x+7-9x=0

$$x ^ { 2 } - 3 x + 7 - 9 x = 0$$

Solve for x

$$x=\sqrt{29}+6\approx 11.385164807$$

$$x=6-\sqrt{29}\approx 0.614835193$$

Steps Using the Quadratic Formula

Combine -3x and -9x to get -12x.

$$x^{2}-12x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 7}}{2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-\left(-12\right)±\sqrt{144-28}}{2}$$

Add 144 to -28.

$$x=\frac{-\left(-12\right)±\sqrt{116}}{2}$$

Take the square root of 116.

$$x=\frac{-\left(-12\right)±2\sqrt{29}}{2}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{29}}{2}$$

Now solve the equation x=\frac{12±2\sqrt{29}}{2} when ± is plus. Add 12 to 2\sqrt{29}.

$$x=\frac{2\sqrt{29}+12}{2}$$

Divide 12+2\sqrt{29} by 2.

$$x=\sqrt{29}+6$$

Now solve the equation x=\frac{12±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from 12.

$$x=\frac{12-2\sqrt{29}}{2}$$

Divide 12-2\sqrt{29} by 2.

$$x=6-\sqrt{29}$$

The equation is now solved.

$$x=\sqrt{29}+6 x=6-\sqrt{29}$$

Steps for Completing the Square

Combine -3x and -9x to get -12x.

$$x^{2}-12x+7=0$$

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-12x=-7$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-7+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-7+36$$

Add -7 to 36.

$$x^{2}-12x+36=29$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=29$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{29}$$

Simplify.

$$x-6=\sqrt{29} x-6=-\sqrt{29}$$

Add 6 to both sides of the equation.

$$x=\sqrt{29}+6 x=6-\sqrt{29}$$