Solve 9x^2-126x+397=0

$$9 { x }^{ 2 } -126x+397 = 0$$

Solve for x

$$x=\frac{2\sqrt{11}}{3}+7\approx 9.211083194$$

$$x=-\frac{2\sqrt{11}}{3}+7\approx 4.788916806$$

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.

$$9x^{2}-126x+397=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -126 for b, and 397 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-126\right)±\sqrt{\left(-126\right)^{2}-4\times 9\times 397}}{2\times 9}$$

Square -126.

$$x=\frac{-\left(-126\right)±\sqrt{15876-4\times 9\times 397}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-126\right)±\sqrt{15876-36\times 397}}{2\times 9}$$

Multiply -36 times 397.

$$x=\frac{-\left(-126\right)±\sqrt{15876-14292}}{2\times 9}$$

Add 15876 to -14292.

$$x=\frac{-\left(-126\right)±\sqrt{1584}}{2\times 9}$$

Take the square root of 1584.

$$x=\frac{-\left(-126\right)±12\sqrt{11}}{2\times 9}$$

The opposite of -126 is 126.

$$x=\frac{126±12\sqrt{11}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{126±12\sqrt{11}}{18}$$

Now solve the equation x=\frac{126±12\sqrt{11}}{18} when ± is plus. Add 126 to 12\sqrt{11}.

$$x=\frac{12\sqrt{11}+126}{18}$$

Divide 126+12\sqrt{11} by 18.

$$x=\frac{2\sqrt{11}}{3}+7$$

Now solve the equation x=\frac{126±12\sqrt{11}}{18} when ± is minus. Subtract 12\sqrt{11} from 126.

$$x=\frac{126-12\sqrt{11}}{18}$$

Divide 126-12\sqrt{11} by 18.

$$x=-\frac{2\sqrt{11}}{3}+7$$

The equation is now solved.

$$x=\frac{2\sqrt{11}}{3}+7 x=-\frac{2\sqrt{11}}{3}+7$$

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.

$$9x^{2}-126x+397=0$$

Subtract 397 from both sides of the equation.

$$9x^{2}-126x+397-397=-397$$

Subtracting 397 from itself leaves 0.

$$9x^{2}-126x=-397$$

Divide both sides by 9.

$$\frac{9x^{2}-126x}{9}=-\frac{397}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{126}{9}\right)x=-\frac{397}{9}$$

Divide -126 by 9.

$$x^{2}-14x=-\frac{397}{9}$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-\frac{397}{9}+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-\frac{397}{9}+49$$

Add -\frac{397}{9} to 49.

$$x^{2}-14x+49=\frac{44}{9}$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=\frac{44}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{\frac{44}{9}}$$

Simplify.

$$x-7=\frac{2\sqrt{11}}{3} x-7=-\frac{2\sqrt{11}}{3}$$

Add 7 to both sides of the equation.

$$x=\frac{2\sqrt{11}}{3}+7 x=-\frac{2\sqrt{11}}{3}+7$$