Solve 3=2x^2-5-60xleft(x-2right)

$$3 = 2 { x }^{ 2 } -5-60x \left( x-2 \right)$$

Solve for x

$$x=2$$

$$x=\frac{2}{29}\approx 0.068965517$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-5-60x\left(x-2\right)=3$$

Subtract 3 from both sides.

$$2x^{2}-5-60x\left(x-2\right)-3=0$$

Use the distributive property to multiply -60x by x-2.

$$2x^{2}-5-60x^{2}+120x-3=0$$

Combine 2x^{2} and -60x^{2} to get -58x^{2}.

$$-58x^{2}-5+120x-3=0$$

Subtract 3 from -5 to get -8.

$$-58x^{2}-8+120x=0$$

Divide both sides by 2.

$$-29x^{2}-4+60x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-29x^{2}+60x-4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -29x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=60 ab=-29\left(-4\right)=116$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 116.

$$1,116 2,58 4,29$$

Calculate the sum for each pair.

$$1+116=117 2+58=60 4+29=33$$

The solution is the pair that gives sum 60.

$$a=58 b=2$$

Rewrite -29x^{2}+60x-4 as \left(-29x^{2}+58x\right)+\left(2x-4\right).

$$\left(-29x^{2}+58x\right)+\left(2x-4\right)$$

Factor out 29x in the first and -2 in the second group.

$$29x\left(-x+2\right)-2\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(29x-2\right)$$

To find equation solutions, solve -x+2=0 and 29x-2=0.

$$x=2 x=\frac{2}{29}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-5-60x\left(x-2\right)=3$$

Subtract 3 from both sides.

$$2x^{2}-5-60x\left(x-2\right)-3=0$$

Use the distributive property to multiply -60x by x-2.

$$2x^{2}-5-60x^{2}+120x-3=0$$

Combine 2x^{2} and -60x^{2} to get -58x^{2}.

$$-58x^{2}-5+120x-3=0$$

Subtract 3 from -5 to get -8.

$$-58x^{2}-8+120x=0$$

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.

$$-58x^{2}+120x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -58 for a, 120 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-58\right)\left(-8\right)}}{2\left(-58\right)}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\left(-58\right)\left(-8\right)}}{2\left(-58\right)}$$

Multiply -4 times -58.

$$x=\frac{-120±\sqrt{14400+232\left(-8\right)}}{2\left(-58\right)}$$

Multiply 232 times -8.

$$x=\frac{-120±\sqrt{14400-1856}}{2\left(-58\right)}$$

Add 14400 to -1856.

$$x=\frac{-120±\sqrt{12544}}{2\left(-58\right)}$$

Take the square root of 12544.

$$x=\frac{-120±112}{2\left(-58\right)}$$

Multiply 2 times -58.

$$x=\frac{-120±112}{-116}$$

Now solve the equation x=\frac{-120±112}{-116} when ± is plus. Add -120 to 112.

$$x=-\frac{8}{-116}$$

Reduce the fraction \frac{-8}{-116} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{29}$$

Now solve the equation x=\frac{-120±112}{-116} when ± is minus. Subtract 112 from -120.

$$x=-\frac{232}{-116}$$

Divide -232 by -116.

$$x=2$$

The equation is now solved.

$$x=\frac{2}{29} x=2$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-5-60x\left(x-2\right)=3$$

Use the distributive property to multiply -60x by x-2.

$$2x^{2}-5-60x^{2}+120x=3$$

Combine 2x^{2} and -60x^{2} to get -58x^{2}.

$$-58x^{2}-5+120x=3$$

Add 5 to both sides.

$$-58x^{2}+120x=3+5$$

Add 3 and 5 to get 8.

$$-58x^{2}+120x=8$$

Divide both sides by -58.

$$\frac{-58x^{2}+120x}{-58}=\frac{8}{-58}$$

Dividing by -58 undoes the multiplication by -58.

$$x^{2}+\frac{120}{-58}x=\frac{8}{-58}$$

Reduce the fraction \frac{120}{-58} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{60}{29}x=\frac{8}{-58}$$

Reduce the fraction \frac{8}{-58} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{60}{29}x=-\frac{4}{29}$$

Divide -\frac{60}{29}, the coefficient of the x term, by 2 to get -\frac{30}{29}. Then add the square of -\frac{30}{29} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{60}{29}x+\left(-\frac{30}{29}\right)^{2}=-\frac{4}{29}+\left(-\frac{30}{29}\right)^{2}$$

Square -\frac{30}{29} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{60}{29}x+\frac{900}{841}=-\frac{4}{29}+\frac{900}{841}$$

Add -\frac{4}{29} to \frac{900}{841} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{60}{29}x+\frac{900}{841}=\frac{784}{841}$$

Factor x^{2}-\frac{60}{29}x+\frac{900}{841}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{30}{29}\right)^{2}=\frac{784}{841}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{30}{29}\right)^{2}}=\sqrt{\frac{784}{841}}$$

Simplify.

$$x-\frac{30}{29}=\frac{28}{29} x-\frac{30}{29}=-\frac{28}{29}$$

Add \frac{30}{29} to both sides of the equation.

$$x=2 x=\frac{2}{29}$$