Solve (5x-7)^2-5(2x+1)(x-2)=-x^2-[-(3x+1)]

$$( 5 x - 7 ) ^ { 2 } - 5 ( 2 x + 1 ) ( x - 2 ) = - x ^ { 2 } - [ - ( 3 x + 1 ) ]$$

Solve for x (complex solution)

$$x=\frac{29+\sqrt{87}i}{16}\approx 1.8125+0.582961191i$$

$$x=\frac{-\sqrt{87}i+29}{16}\approx 1.8125-0.582961191i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-7\right)^{2}.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-\left(3x+1\right)\right)$$

To find the opposite of 3x+1, find the opposite of each term.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-3x-1\right)$$

To find the opposite of -3x-1, find the opposite of each term.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}+3x+1$$

Add x^{2} to both sides.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}=3x+1$$

Subtract 3x from both sides.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}-3x=1$$

Subtract 1 from both sides.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}-3x-1=0$$

Use the distributive property to multiply -5 by 2x+1.

$$25x^{2}-70x+49+\left(-10x-5\right)\left(x-2\right)+x^{2}-3x-1=0$$

Use the distributive property to multiply -10x-5 by x-2 and combine like terms.

$$25x^{2}-70x+49-10x^{2}+15x+10+x^{2}-3x-1=0$$

Combine 25x^{2} and -10x^{2} to get 15x^{2}.

$$15x^{2}-70x+49+15x+10+x^{2}-3x-1=0$$

Combine -70x and 15x to get -55x.

$$15x^{2}-55x+49+10+x^{2}-3x-1=0$$

Add 49 and 10 to get 59.

$$15x^{2}-55x+59+x^{2}-3x-1=0$$

Combine 15x^{2} and x^{2} to get 16x^{2}.

$$16x^{2}-55x+59-3x-1=0$$

Combine -55x and -3x to get -58x.

$$16x^{2}-58x+59-1=0$$

Subtract 1 from 59 to get 58.

$$16x^{2}-58x+58=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -58 for b, and 58 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-58\right)±\sqrt{\left(-58\right)^{2}-4\times 16\times 58}}{2\times 16}$$

Square -58.

$$x=\frac{-\left(-58\right)±\sqrt{3364-4\times 16\times 58}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-58\right)±\sqrt{3364-64\times 58}}{2\times 16}$$

Multiply -64 times 58.

$$x=\frac{-\left(-58\right)±\sqrt{3364-3712}}{2\times 16}$$

Add 3364 to -3712.

$$x=\frac{-\left(-58\right)±\sqrt{-348}}{2\times 16}$$

Take the square root of -348.

$$x=\frac{-\left(-58\right)±2\sqrt{87}i}{2\times 16}$$

The opposite of -58 is 58.

$$x=\frac{58±2\sqrt{87}i}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{58±2\sqrt{87}i}{32}$$

Now solve the equation x=\frac{58±2\sqrt{87}i}{32} when ± is plus. Add 58 to 2i\sqrt{87}.

$$x=\frac{58+2\sqrt{87}i}{32}$$

Divide 58+2i\sqrt{87} by 32.

$$x=\frac{29+\sqrt{87}i}{16}$$

Now solve the equation x=\frac{58±2\sqrt{87}i}{32} when ± is minus. Subtract 2i\sqrt{87} from 58.

$$x=\frac{-2\sqrt{87}i+58}{32}$$

Divide 58-2i\sqrt{87} by 32.

$$x=\frac{-\sqrt{87}i+29}{16}$$

The equation is now solved.

$$x=\frac{29+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i+29}{16}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-7\right)^{2}.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-\left(3x+1\right)\right)$$

To find the opposite of 3x+1, find the opposite of each term.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}-\left(-3x-1\right)$$

To find the opposite of -3x-1, find the opposite of each term.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)=-x^{2}+3x+1$$

Add x^{2} to both sides.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}=3x+1$$

Subtract 3x from both sides.

$$25x^{2}-70x+49-5\left(2x+1\right)\left(x-2\right)+x^{2}-3x=1$$

Use the distributive property to multiply -5 by 2x+1.

$$25x^{2}-70x+49+\left(-10x-5\right)\left(x-2\right)+x^{2}-3x=1$$

Use the distributive property to multiply -10x-5 by x-2 and combine like terms.

$$25x^{2}-70x+49-10x^{2}+15x+10+x^{2}-3x=1$$

Combine 25x^{2} and -10x^{2} to get 15x^{2}.

$$15x^{2}-70x+49+15x+10+x^{2}-3x=1$$

Combine -70x and 15x to get -55x.

$$15x^{2}-55x+49+10+x^{2}-3x=1$$

Add 49 and 10 to get 59.

$$15x^{2}-55x+59+x^{2}-3x=1$$

Combine 15x^{2} and x^{2} to get 16x^{2}.

$$16x^{2}-55x+59-3x=1$$

Combine -55x and -3x to get -58x.

$$16x^{2}-58x+59=1$$

Subtract 59 from both sides.

$$16x^{2}-58x=1-59$$

Subtract 59 from 1 to get -58.

$$16x^{2}-58x=-58$$

Divide both sides by 16.

$$\frac{16x^{2}-58x}{16}=-\frac{58}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{58}{16}\right)x=-\frac{58}{16}$$

Reduce the fraction \frac{-58}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{29}{8}x=-\frac{58}{16}$$

Reduce the fraction \frac{-58}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{29}{8}x=-\frac{29}{8}$$

Divide -\frac{29}{8}, the coefficient of the x term, by 2 to get -\frac{29}{16}. Then add the square of -\frac{29}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{29}{8}x+\left(-\frac{29}{16}\right)^{2}=-\frac{29}{8}+\left(-\frac{29}{16}\right)^{2}$$

Square -\frac{29}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{29}{8}x+\frac{841}{256}=-\frac{29}{8}+\frac{841}{256}$$

Add -\frac{29}{8} to \frac{841}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{29}{8}x+\frac{841}{256}=-\frac{87}{256}$$

Factor x^{2}-\frac{29}{8}x+\frac{841}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{16}\right)^{2}=-\frac{87}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{16}\right)^{2}}=\sqrt{-\frac{87}{256}}$$

Simplify.

$$x-\frac{29}{16}=\frac{\sqrt{87}i}{16} x-\frac{29}{16}=-\frac{\sqrt{87}i}{16}$$

Add \frac{29}{16} to both sides of the equation.

$$x=\frac{29+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i+29}{16}$$