Solve [2-(2x+5)+x-1/3]+5x^2=x^2-12x-x-2/5

$$[ 2 - ( 2 x + 5 ) + \frac { x - 1 } { 3 } ] + 5 x ^ { 2 } = x ^ { 2 } - 12 x - \frac { x - 2 } { 5 }$$

Solve for x

$$x=\frac{\sqrt{9601}-79}{60}\approx 0.316411545$$

$$x=\frac{-\sqrt{9601}-79}{60}\approx -2.949744878$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$30-15\left(2x+5\right)+5\left(x-1\right)+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Use the distributive property to multiply -15 by 2x+5.

$$30-30x-75+5\left(x-1\right)+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Subtract 75 from 30 to get -45.

$$-45-30x+5\left(x-1\right)+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

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

$$-45-30x+5x-5+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Combine -30x and 5x to get -25x.

$$-45-25x-5+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Subtract 5 from -45 to get -50.

$$-50-25x+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Use the distributive property to multiply -3 by x-2.

$$-50-25x+75x^{2}=15x^{2}-180x-3x+6$$

Combine -180x and -3x to get -183x.

$$-50-25x+75x^{2}=15x^{2}-183x+6$$

Subtract 15x^{2} from both sides.

$$-50-25x+75x^{2}-15x^{2}=-183x+6$$

Combine 75x^{2} and -15x^{2} to get 60x^{2}.

$$-50-25x+60x^{2}=-183x+6$$

Add 183x to both sides.

$$-50-25x+60x^{2}+183x=6$$

Combine -25x and 183x to get 158x.

$$-50+158x+60x^{2}=6$$

Subtract 6 from both sides.

$$-50+158x+60x^{2}-6=0$$

Subtract 6 from -50 to get -56.

$$-56+158x+60x^{2}=0$$

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.

$$60x^{2}+158x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 60 for a, 158 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-158±\sqrt{158^{2}-4\times 60\left(-56\right)}}{2\times 60}$$

Square 158.

$$x=\frac{-158±\sqrt{24964-4\times 60\left(-56\right)}}{2\times 60}$$

Multiply -4 times 60.

$$x=\frac{-158±\sqrt{24964-240\left(-56\right)}}{2\times 60}$$

Multiply -240 times -56.

$$x=\frac{-158±\sqrt{24964+13440}}{2\times 60}$$

Add 24964 to 13440.

$$x=\frac{-158±\sqrt{38404}}{2\times 60}$$

Take the square root of 38404.

$$x=\frac{-158±2\sqrt{9601}}{2\times 60}$$

Multiply 2 times 60.

$$x=\frac{-158±2\sqrt{9601}}{120}$$

Now solve the equation x=\frac{-158±2\sqrt{9601}}{120} when ± is plus. Add -158 to 2\sqrt{9601}.

$$x=\frac{2\sqrt{9601}-158}{120}$$

Divide -158+2\sqrt{9601} by 120.

$$x=\frac{\sqrt{9601}-79}{60}$$

Now solve the equation x=\frac{-158±2\sqrt{9601}}{120} when ± is minus. Subtract 2\sqrt{9601} from -158.

$$x=\frac{-2\sqrt{9601}-158}{120}$$

Divide -158-2\sqrt{9601} by 120.

$$x=\frac{-\sqrt{9601}-79}{60}$$

The equation is now solved.

$$x=\frac{\sqrt{9601}-79}{60} x=\frac{-\sqrt{9601}-79}{60}$$

Steps for Completing the Square

Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$30-15\left(2x+5\right)+5\left(x-1\right)+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Use the distributive property to multiply -15 by 2x+5.

$$30-30x-75+5\left(x-1\right)+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Subtract 75 from 30 to get -45.

$$-45-30x+5\left(x-1\right)+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

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

$$-45-30x+5x-5+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Combine -30x and 5x to get -25x.

$$-45-25x-5+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Subtract 5 from -45 to get -50.

$$-50-25x+75x^{2}=15x^{2}-180x-3\left(x-2\right)$$

Use the distributive property to multiply -3 by x-2.

$$-50-25x+75x^{2}=15x^{2}-180x-3x+6$$

Combine -180x and -3x to get -183x.

$$-50-25x+75x^{2}=15x^{2}-183x+6$$

Subtract 15x^{2} from both sides.

$$-50-25x+75x^{2}-15x^{2}=-183x+6$$

Combine 75x^{2} and -15x^{2} to get 60x^{2}.

$$-50-25x+60x^{2}=-183x+6$$

Add 183x to both sides.

$$-50-25x+60x^{2}+183x=6$$

Combine -25x and 183x to get 158x.

$$-50+158x+60x^{2}=6$$

Add 50 to both sides.

$$158x+60x^{2}=6+50$$

Add 6 and 50 to get 56.

$$158x+60x^{2}=56$$

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.

$$60x^{2}+158x=56$$

Divide both sides by 60.

$$\frac{60x^{2}+158x}{60}=\frac{56}{60}$$

Dividing by 60 undoes the multiplication by 60.

$$x^{2}+\frac{158}{60}x=\frac{56}{60}$$

Reduce the fraction \frac{158}{60} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{79}{30}x=\frac{56}{60}$$

Reduce the fraction \frac{56}{60} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{79}{30}x=\frac{14}{15}$$

Divide \frac{79}{30}, the coefficient of the x term, by 2 to get \frac{79}{60}. Then add the square of \frac{79}{60} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{79}{30}x+\left(\frac{79}{60}\right)^{2}=\frac{14}{15}+\left(\frac{79}{60}\right)^{2}$$

Square \frac{79}{60} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{79}{30}x+\frac{6241}{3600}=\frac{14}{15}+\frac{6241}{3600}$$

Add \frac{14}{15} to \frac{6241}{3600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{79}{30}x+\frac{6241}{3600}=\frac{9601}{3600}$$

Factor x^{2}+\frac{79}{30}x+\frac{6241}{3600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{79}{60}\right)^{2}=\frac{9601}{3600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{79}{60}\right)^{2}}=\sqrt{\frac{9601}{3600}}$$

Simplify.

$$x+\frac{79}{60}=\frac{\sqrt{9601}}{60} x+\frac{79}{60}=-\frac{\sqrt{9601}}{60}$$

Subtract \frac{79}{60} from both sides of the equation.

$$x=\frac{\sqrt{9601}-79}{60} x=\frac{-\sqrt{9601}-79}{60}$$