Solve 3=7x(2x-5)-x(2x-5)

$$3 = 7 x ( 2 x - 5 ) - x ( 2 x - 5 )$$

Solve for x

$$x = \frac{\sqrt{29} + 5}{4} \approx 2.596291202$$

$$x=\frac{5-\sqrt{29}}{4}\approx -0.096291202$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7x by 2x-5.

$$3=14x^{2}-35x-x\left(2x-5\right)$$

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

$$3=14x^{2}-35x-\left(2x^{2}-5x\right)$$

To find the opposite of 2x^{2}-5x, find the opposite of each term.

$$3=14x^{2}-35x-2x^{2}-\left(-5x\right)$$

The opposite of -5x is 5x.

$$3=14x^{2}-35x-2x^{2}+5x$$

Combine 14x^{2} and -2x^{2} to get 12x^{2}.

$$3=12x^{2}-35x+5x$$

Combine -35x and 5x to get -30x.

$$3=12x^{2}-30x$$

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

$$12x^{2}-30x=3$$

Subtract 3 from both sides.

$$12x^{2}-30x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -30 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 12\left(-3\right)}}{2\times 12}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 12\left(-3\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-30\right)±\sqrt{900-48\left(-3\right)}}{2\times 12}$$

Multiply -48 times -3.

$$x=\frac{-\left(-30\right)±\sqrt{900+144}}{2\times 12}$$

Add 900 to 144.

$$x=\frac{-\left(-30\right)±\sqrt{1044}}{2\times 12}$$

Take the square root of 1044.

$$x=\frac{-\left(-30\right)±6\sqrt{29}}{2\times 12}$$

The opposite of -30 is 30.

$$x=\frac{30±6\sqrt{29}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{30±6\sqrt{29}}{24}$$

Now solve the equation x=\frac{30±6\sqrt{29}}{24} when ± is plus. Add 30 to 6\sqrt{29}.

$$x=\frac{6\sqrt{29}+30}{24}$$

Divide 30+6\sqrt{29} by 24.

$$x=\frac{\sqrt{29}+5}{4}$$

Now solve the equation x=\frac{30±6\sqrt{29}}{24} when ± is minus. Subtract 6\sqrt{29} from 30.

$$x=\frac{30-6\sqrt{29}}{24}$$

Divide 30-6\sqrt{29} by 24.

$$x=\frac{5-\sqrt{29}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{29}+5}{4} x=\frac{5-\sqrt{29}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 7x by 2x-5.

$$3=14x^{2}-35x-x\left(2x-5\right)$$

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

$$3=14x^{2}-35x-\left(2x^{2}-5x\right)$$

To find the opposite of 2x^{2}-5x, find the opposite of each term.

$$3=14x^{2}-35x-2x^{2}-\left(-5x\right)$$

The opposite of -5x is 5x.

$$3=14x^{2}-35x-2x^{2}+5x$$

Combine 14x^{2} and -2x^{2} to get 12x^{2}.

$$3=12x^{2}-35x+5x$$

Combine -35x and 5x to get -30x.

$$3=12x^{2}-30x$$

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

$$12x^{2}-30x=3$$

Divide both sides by 12.

$$\frac{12x^{2}-30x}{12}=\frac{3}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{30}{12}\right)x=\frac{3}{12}$$

Reduce the fraction \frac{-30}{12} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{5}{2}x=\frac{3}{12}$$

Reduce the fraction \frac{3}{12} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{5}{2}x=\frac{1}{4}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=\frac{1}{4}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{1}{4}+\frac{25}{16}$$

Add \frac{1}{4} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{29}{16}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{29}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{29}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{\sqrt{29}}{4} x-\frac{5}{4}=-\frac{\sqrt{29}}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{\sqrt{29}+5}{4} x=\frac{5-\sqrt{29}}{4}$$