Solve 11x^2-34x+3=0

$$11 { x }^{ 2 } -34x+3=0$$

Solve for x

$$x=\frac{1}{11}\approx 0.090909091$$

$$x=3$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 11x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=-34 ab=11\times 3=33$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 33.

$$-1,-33 -3,-11$$

Calculate the sum for each pair.

$$-1-33=-34 -3-11=-14$$

The solution is the pair that gives sum -34.

$$a=-33 b=-1$$

Rewrite 11x^{2}-34x+3 as \left(11x^{2}-33x\right)+\left(-x+3\right).

$$\left(11x^{2}-33x\right)+\left(-x+3\right)$$

Factor out 11x in the first and -1 in the second group.

$$11x\left(x-3\right)-\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(11x-1\right)$$

To find equation solutions, solve x-3=0 and 11x-1=0.

$$x=3 x=\frac{1}{11}$$

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.

$$11x^{2}-34x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, -34 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 11\times 3}}{2\times 11}$$

Square -34.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 11\times 3}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-34\right)±\sqrt{1156-44\times 3}}{2\times 11}$$

Multiply -44 times 3.

$$x=\frac{-\left(-34\right)±\sqrt{1156-132}}{2\times 11}$$

Add 1156 to -132.

$$x=\frac{-\left(-34\right)±\sqrt{1024}}{2\times 11}$$

Take the square root of 1024.

$$x=\frac{-\left(-34\right)±32}{2\times 11}$$

The opposite of -34 is 34.

$$x=\frac{34±32}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{34±32}{22}$$

Now solve the equation x=\frac{34±32}{22} when ± is plus. Add 34 to 32.

$$x=\frac{66}{22}$$

Divide 66 by 22.

$$x=3$$

Now solve the equation x=\frac{34±32}{22} when ± is minus. Subtract 32 from 34.

$$x=\frac{2}{22}$$

Reduce the fraction \frac{2}{22} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{11}$$

The equation is now solved.

$$x=3 x=\frac{1}{11}$$

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.

$$11x^{2}-34x+3=0$$

Subtract 3 from both sides of the equation.

$$11x^{2}-34x+3-3=-3$$

Subtracting 3 from itself leaves 0.

$$11x^{2}-34x=-3$$

Divide both sides by 11.

$$\frac{11x^{2}-34x}{11}=-\frac{3}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}-\frac{34}{11}x=-\frac{3}{11}$$

Divide -\frac{34}{11}, the coefficient of the x term, by 2 to get -\frac{17}{11}. Then add the square of -\frac{17}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{34}{11}x+\left(-\frac{17}{11}\right)^{2}=-\frac{3}{11}+\left(-\frac{17}{11}\right)^{2}$$

Square -\frac{17}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{34}{11}x+\frac{289}{121}=-\frac{3}{11}+\frac{289}{121}$$

Add -\frac{3}{11} to \frac{289}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{34}{11}x+\frac{289}{121}=\frac{256}{121}$$

Factor x^{2}-\frac{34}{11}x+\frac{289}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{11}\right)^{2}=\frac{256}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{11}\right)^{2}}=\sqrt{\frac{256}{121}}$$

Simplify.

$$x-\frac{17}{11}=\frac{16}{11} x-\frac{17}{11}=-\frac{16}{11}$$

Add \frac{17}{11} to both sides of the equation.

$$x=3 x=\frac{1}{11}$$