Solve (5x-2)^2-(2x-1)(2x+1)=47+x

$$( 5 x - 2 ) ^ { 2 } - ( 2 x - 1 ) ( 2 x + 1 ) = 47 + x$$

Solve for x

$$x=-1$$

$$x=2$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-2\right)^{2}.

$$25x^{2}-20x+4-\left(2x-1\right)\left(2x+1\right)=47+x$$

Consider \left(2x-1\right)\left(2x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$25x^{2}-20x+4-\left(\left(2x\right)^{2}-1\right)=47+x$$

Expand \left(2x\right)^{2}.

$$25x^{2}-20x+4-\left(2^{2}x^{2}-1\right)=47+x$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-20x+4-\left(4x^{2}-1\right)=47+x$$

To find the opposite of 4x^{2}-1, find the opposite of each term.

$$25x^{2}-20x+4-4x^{2}+1=47+x$$

Combine 25x^{2} and -4x^{2} to get 21x^{2}.

$$21x^{2}-20x+4+1=47+x$$

Add 4 and 1 to get 5.

$$21x^{2}-20x+5=47+x$$

Subtract 47 from both sides.

$$21x^{2}-20x+5-47=x$$

Subtract 47 from 5 to get -42.

$$21x^{2}-20x-42=x$$

Subtract x from both sides.

$$21x^{2}-20x-42-x=0$$

Combine -20x and -x to get -21x.

$$21x^{2}-21x-42=0$$

Divide both sides by 21.

$$x^{2}-x-2=0$$

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

$$a+b=-1 ab=1\left(-2\right)=-2$$

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. The only such pair is the system solution.

$$a=-2 b=1$$

Rewrite x^{2}-x-2 as \left(x^{2}-2x\right)+\left(x-2\right).

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

Factor out x in x^{2}-2x.

$$x\left(x-2\right)+x-2$$

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

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

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

$$x=2 x=-1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-2\right)^{2}.

$$25x^{2}-20x+4-\left(2x-1\right)\left(2x+1\right)=47+x$$

Consider \left(2x-1\right)\left(2x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$25x^{2}-20x+4-\left(\left(2x\right)^{2}-1\right)=47+x$$

Expand \left(2x\right)^{2}.

$$25x^{2}-20x+4-\left(2^{2}x^{2}-1\right)=47+x$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-20x+4-\left(4x^{2}-1\right)=47+x$$

To find the opposite of 4x^{2}-1, find the opposite of each term.

$$25x^{2}-20x+4-4x^{2}+1=47+x$$

Combine 25x^{2} and -4x^{2} to get 21x^{2}.

$$21x^{2}-20x+4+1=47+x$$

Add 4 and 1 to get 5.

$$21x^{2}-20x+5=47+x$$

Subtract 47 from both sides.

$$21x^{2}-20x+5-47=x$$

Subtract 47 from 5 to get -42.

$$21x^{2}-20x-42=x$$

Subtract x from both sides.

$$21x^{2}-20x-42-x=0$$

Combine -20x and -x to get -21x.

$$21x^{2}-21x-42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -21 for b, and -42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 21\left(-42\right)}}{2\times 21}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 21\left(-42\right)}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-21\right)±\sqrt{441-84\left(-42\right)}}{2\times 21}$$

Multiply -84 times -42.

$$x=\frac{-\left(-21\right)±\sqrt{441+3528}}{2\times 21}$$

Add 441 to 3528.

$$x=\frac{-\left(-21\right)±\sqrt{3969}}{2\times 21}$$

Take the square root of 3969.

$$x=\frac{-\left(-21\right)±63}{2\times 21}$$

The opposite of -21 is 21.

$$x=\frac{21±63}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{21±63}{42}$$

Now solve the equation x=\frac{21±63}{42} when ± is plus. Add 21 to 63.

$$x=\frac{84}{42}$$

Divide 84 by 42.

$$x=2$$

Now solve the equation x=\frac{21±63}{42} when ± is minus. Subtract 63 from 21.

$$x=-\frac{42}{42}$$

Divide -42 by 42.

$$x=-1$$

The equation is now solved.

$$x=2 x=-1$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-2\right)^{2}.

$$25x^{2}-20x+4-\left(2x-1\right)\left(2x+1\right)=47+x$$

Consider \left(2x-1\right)\left(2x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$25x^{2}-20x+4-\left(\left(2x\right)^{2}-1\right)=47+x$$

Expand \left(2x\right)^{2}.

$$25x^{2}-20x+4-\left(2^{2}x^{2}-1\right)=47+x$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-20x+4-\left(4x^{2}-1\right)=47+x$$

To find the opposite of 4x^{2}-1, find the opposite of each term.

$$25x^{2}-20x+4-4x^{2}+1=47+x$$

Combine 25x^{2} and -4x^{2} to get 21x^{2}.

$$21x^{2}-20x+4+1=47+x$$

Add 4 and 1 to get 5.

$$21x^{2}-20x+5=47+x$$

Subtract x from both sides.

$$21x^{2}-20x+5-x=47$$

Combine -20x and -x to get -21x.

$$21x^{2}-21x+5=47$$

Subtract 5 from both sides.

$$21x^{2}-21x=47-5$$

Subtract 5 from 47 to get 42.

$$21x^{2}-21x=42$$

Divide both sides by 21.

$$\frac{21x^{2}-21x}{21}=\frac{42}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}+\left(-\frac{21}{21}\right)x=\frac{42}{21}$$

Divide -21 by 21.

$$x^{2}-x=\frac{42}{21}$$

Divide 42 by 21.

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

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=2+\frac{1}{4}$$

Add 2 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{9}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{3}{2} x-\frac{1}{2}=-\frac{3}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=2 x=-1$$