Solve 147x^2-63x=54

$$147 { x }^{ 2 } -63x=54$$

Solve for x

$$x=-\frac{3}{7}\approx -0.428571429$$

$$x=\frac{6}{7}\approx 0.857142857$$

Steps Using Factoring By Grouping

Subtract 54 from both sides.

$$147x^{2}-63x-54=0$$

Divide both sides by 3.

$$49x^{2}-21x-18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 49x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=-21 ab=49\left(-18\right)=-882$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -882.

$$1,-882 2,-441 3,-294 6,-147 7,-126 9,-98 14,-63 18,-49 21,-42$$

Calculate the sum for each pair.

$$1-882=-881 2-441=-439 3-294=-291 6-147=-141 7-126=-119 9-98=-89 14-63=-49 18-49=-31 21-42=-21$$

The solution is the pair that gives sum -21.

$$a=-42 b=21$$

Rewrite 49x^{2}-21x-18 as \left(49x^{2}-42x\right)+\left(21x-18\right).

$$\left(49x^{2}-42x\right)+\left(21x-18\right)$$

Factor out 7x in the first and 3 in the second group.

$$7x\left(7x-6\right)+3\left(7x-6\right)$$

Factor out common term 7x-6 by using distributive property.

$$\left(7x-6\right)\left(7x+3\right)$$

To find equation solutions, solve 7x-6=0 and 7x+3=0.

$$x=\frac{6}{7} x=-\frac{3}{7}$$

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.

$$147x^{2}-63x=54$$

Subtract 54 from both sides of the equation.

$$147x^{2}-63x-54=54-54$$

Subtracting 54 from itself leaves 0.

$$147x^{2}-63x-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 147 for a, -63 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-63\right)±\sqrt{\left(-63\right)^{2}-4\times 147\left(-54\right)}}{2\times 147}$$

Square -63.

$$x=\frac{-\left(-63\right)±\sqrt{3969-4\times 147\left(-54\right)}}{2\times 147}$$

Multiply -4 times 147.

$$x=\frac{-\left(-63\right)±\sqrt{3969-588\left(-54\right)}}{2\times 147}$$

Multiply -588 times -54.

$$x=\frac{-\left(-63\right)±\sqrt{3969+31752}}{2\times 147}$$

Add 3969 to 31752.

$$x=\frac{-\left(-63\right)±\sqrt{35721}}{2\times 147}$$

Take the square root of 35721.

$$x=\frac{-\left(-63\right)±189}{2\times 147}$$

The opposite of -63 is 63.

$$x=\frac{63±189}{2\times 147}$$

Multiply 2 times 147.

$$x=\frac{63±189}{294}$$

Now solve the equation x=\frac{63±189}{294} when ± is plus. Add 63 to 189.

$$x=\frac{252}{294}$$

Reduce the fraction \frac{252}{294} to lowest terms by extracting and canceling out 42.

$$x=\frac{6}{7}$$

Now solve the equation x=\frac{63±189}{294} when ± is minus. Subtract 189 from 63.

$$x=-\frac{126}{294}$$

Reduce the fraction \frac{-126}{294} to lowest terms by extracting and canceling out 42.

$$x=-\frac{3}{7}$$

The equation is now solved.

$$x=\frac{6}{7} x=-\frac{3}{7}$$

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.

$$147x^{2}-63x=54$$

Divide both sides by 147.

$$\frac{147x^{2}-63x}{147}=\frac{54}{147}$$

Dividing by 147 undoes the multiplication by 147.

$$x^{2}+\left(-\frac{63}{147}\right)x=\frac{54}{147}$$

Reduce the fraction \frac{-63}{147} to lowest terms by extracting and canceling out 21.

$$x^{2}-\frac{3}{7}x=\frac{54}{147}$$

Reduce the fraction \frac{54}{147} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{3}{7}x=\frac{18}{49}$$

Divide -\frac{3}{7}, the coefficient of the x term, by 2 to get -\frac{3}{14}. Then add the square of -\frac{3}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{7}x+\left(-\frac{3}{14}\right)^{2}=\frac{18}{49}+\left(-\frac{3}{14}\right)^{2}$$

Square -\frac{3}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{18}{49}+\frac{9}{196}$$

Add \frac{18}{49} to \frac{9}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{81}{196}$$

Factor x^{2}-\frac{3}{7}x+\frac{9}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{14}\right)^{2}=\frac{81}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{14}\right)^{2}}=\sqrt{\frac{81}{196}}$$

Simplify.

$$x-\frac{3}{14}=\frac{9}{14} x-\frac{3}{14}=-\frac{9}{14}$$

Add \frac{3}{14} to both sides of the equation.

$$x=\frac{6}{7} x=-\frac{3}{7}$$