Solve (5x-2)(7x-4)=(3x+4)(7x+3)-5x

$$(5x-2)(7x-4)=(3x+4)(7x+3)-5x$$

Solve for x

$$x = \frac{\sqrt{1145} + 33}{14} \approx 4.774132045$$

$$x=\frac{33-\sqrt{1145}}{14}\approx -0.059846331$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x-2 by 7x-4 and combine like terms.

$$35x^{2}-34x+8=\left(3x+4\right)\left(7x+3\right)-5x$$

Use the distributive property to multiply 3x+4 by 7x+3 and combine like terms.

$$35x^{2}-34x+8=21x^{2}+37x+12-5x$$

Combine 37x and -5x to get 32x.

$$35x^{2}-34x+8=21x^{2}+32x+12$$

Subtract 21x^{2} from both sides.

$$35x^{2}-34x+8-21x^{2}=32x+12$$

Combine 35x^{2} and -21x^{2} to get 14x^{2}.

$$14x^{2}-34x+8=32x+12$$

Subtract 32x from both sides.

$$14x^{2}-34x+8-32x=12$$

Combine -34x and -32x to get -66x.

$$14x^{2}-66x+8=12$$

Subtract 12 from both sides.

$$14x^{2}-66x+8-12=0$$

Subtract 12 from 8 to get -4.

$$14x^{2}-66x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, -66 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 14\left(-4\right)}}{2\times 14}$$

Square -66.

$$x=\frac{-\left(-66\right)±\sqrt{4356-4\times 14\left(-4\right)}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-\left(-66\right)±\sqrt{4356-56\left(-4\right)}}{2\times 14}$$

Multiply -56 times -4.

$$x=\frac{-\left(-66\right)±\sqrt{4356+224}}{2\times 14}$$

Add 4356 to 224.

$$x=\frac{-\left(-66\right)±\sqrt{4580}}{2\times 14}$$

Take the square root of 4580.

$$x=\frac{-\left(-66\right)±2\sqrt{1145}}{2\times 14}$$

The opposite of -66 is 66.

$$x=\frac{66±2\sqrt{1145}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{66±2\sqrt{1145}}{28}$$

Now solve the equation x=\frac{66±2\sqrt{1145}}{28} when ± is plus. Add 66 to 2\sqrt{1145}.

$$x=\frac{2\sqrt{1145}+66}{28}$$

Divide 66+2\sqrt{1145} by 28.

$$x=\frac{\sqrt{1145}+33}{14}$$

Now solve the equation x=\frac{66±2\sqrt{1145}}{28} when ± is minus. Subtract 2\sqrt{1145} from 66.

$$x=\frac{66-2\sqrt{1145}}{28}$$

Divide 66-2\sqrt{1145} by 28.

$$x=\frac{33-\sqrt{1145}}{14}$$

The equation is now solved.

$$x=\frac{\sqrt{1145}+33}{14} x=\frac{33-\sqrt{1145}}{14}$$

Steps for Completing the Square

Use the distributive property to multiply 5x-2 by 7x-4 and combine like terms.

$$35x^{2}-34x+8=\left(3x+4\right)\left(7x+3\right)-5x$$

Use the distributive property to multiply 3x+4 by 7x+3 and combine like terms.

$$35x^{2}-34x+8=21x^{2}+37x+12-5x$$

Combine 37x and -5x to get 32x.

$$35x^{2}-34x+8=21x^{2}+32x+12$$

Subtract 21x^{2} from both sides.

$$35x^{2}-34x+8-21x^{2}=32x+12$$

Combine 35x^{2} and -21x^{2} to get 14x^{2}.

$$14x^{2}-34x+8=32x+12$$

Subtract 32x from both sides.

$$14x^{2}-34x+8-32x=12$$

Combine -34x and -32x to get -66x.

$$14x^{2}-66x+8=12$$

Subtract 8 from both sides.

$$14x^{2}-66x=12-8$$

Subtract 8 from 12 to get 4.

$$14x^{2}-66x=4$$

Divide both sides by 14.

$$\frac{14x^{2}-66x}{14}=\frac{4}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$x^{2}+\left(-\frac{66}{14}\right)x=\frac{4}{14}$$

Reduce the fraction \frac{-66}{14} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{33}{7}x=\frac{4}{14}$$

Reduce the fraction \frac{4}{14} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{33}{7}x=\frac{2}{7}$$

Divide -\frac{33}{7}, the coefficient of the x term, by 2 to get -\frac{33}{14}. Then add the square of -\frac{33}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{33}{7}x+\left(-\frac{33}{14}\right)^{2}=\frac{2}{7}+\left(-\frac{33}{14}\right)^{2}$$

Square -\frac{33}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{33}{7}x+\frac{1089}{196}=\frac{2}{7}+\frac{1089}{196}$$

Add \frac{2}{7} to \frac{1089}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{33}{7}x+\frac{1089}{196}=\frac{1145}{196}$$

Factor x^{2}-\frac{33}{7}x+\frac{1089}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{33}{14}\right)^{2}=\frac{1145}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{33}{14}\right)^{2}}=\sqrt{\frac{1145}{196}}$$

Simplify.

$$x-\frac{33}{14}=\frac{\sqrt{1145}}{14} x-\frac{33}{14}=-\frac{\sqrt{1145}}{14}$$

Add \frac{33}{14} to both sides of the equation.

$$x=\frac{\sqrt{1145}+33}{14} x=\frac{33-\sqrt{1145}}{14}$$