Solve 56x^2-30x+1=0

$$56 { x }^{ 2 } -30x+1=0$$

Solve for x

$$x=\frac{1}{28}\approx 0.035714286$$

$$x=\frac{1}{2}=0.5$$

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 56x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=-30 ab=56\times 1=56$$

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 56.

$$-1,-56 -2,-28 -4,-14 -7,-8$$

Calculate the sum for each pair.

$$-1-56=-57 -2-28=-30 -4-14=-18 -7-8=-15$$

The solution is the pair that gives sum -30.

$$a=-28 b=-2$$

Rewrite 56x^{2}-30x+1 as \left(56x^{2}-28x\right)+\left(-2x+1\right).

$$\left(56x^{2}-28x\right)+\left(-2x+1\right)$$

Factor out 28x in the first and -1 in the second group.

$$28x\left(2x-1\right)-\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(28x-1\right)$$

To find equation solutions, solve 2x-1=0 and 28x-1=0.

$$x=\frac{1}{2} x=\frac{1}{28}$$

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.

$$56x^{2}-30x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 56 for a, -30 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 56}}{2\times 56}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 56}}{2\times 56}$$

Multiply -4 times 56.

$$x=\frac{-\left(-30\right)±\sqrt{900-224}}{2\times 56}$$

Add 900 to -224.

$$x=\frac{-\left(-30\right)±\sqrt{676}}{2\times 56}$$

Take the square root of 676.

$$x=\frac{-\left(-30\right)±26}{2\times 56}$$

The opposite of -30 is 30.

$$x=\frac{30±26}{2\times 56}$$

Multiply 2 times 56.

$$x=\frac{30±26}{112}$$

Now solve the equation x=\frac{30±26}{112} when ± is plus. Add 30 to 26.

$$x=\frac{56}{112}$$

Reduce the fraction \frac{56}{112} to lowest terms by extracting and canceling out 56.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{30±26}{112} when ± is minus. Subtract 26 from 30.

$$x=\frac{4}{112}$$

Reduce the fraction \frac{4}{112} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{28}$$

The equation is now solved.

$$x=\frac{1}{2} x=\frac{1}{28}$$

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.

$$56x^{2}-30x+1=0$$

Subtract 1 from both sides of the equation.

$$56x^{2}-30x+1-1=-1$$

Subtracting 1 from itself leaves 0.

$$56x^{2}-30x=-1$$

Divide both sides by 56.

$$\frac{56x^{2}-30x}{56}=-\frac{1}{56}$$

Dividing by 56 undoes the multiplication by 56.

$$x^{2}+\left(-\frac{30}{56}\right)x=-\frac{1}{56}$$

Reduce the fraction \frac{-30}{56} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{15}{28}x=-\frac{1}{56}$$

Divide -\frac{15}{28}, the coefficient of the x term, by 2 to get -\frac{15}{56}. Then add the square of -\frac{15}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{28}x+\left(-\frac{15}{56}\right)^{2}=-\frac{1}{56}+\left(-\frac{15}{56}\right)^{2}$$

Square -\frac{15}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{28}x+\frac{225}{3136}=-\frac{1}{56}+\frac{225}{3136}$$

Add -\frac{1}{56} to \frac{225}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{28}x+\frac{225}{3136}=\frac{169}{3136}$$

Factor x^{2}-\frac{15}{28}x+\frac{225}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{56}\right)^{2}=\frac{169}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{56}\right)^{2}}=\sqrt{\frac{169}{3136}}$$

Simplify.

$$x-\frac{15}{56}=\frac{13}{56} x-\frac{15}{56}=-\frac{13}{56}$$

Add \frac{15}{56} to both sides of the equation.

$$x=\frac{1}{2} x=\frac{1}{28}$$