Solve 1-3x/2(4x+3)^2=1/49

$$\frac { 1 - 3 x } { 2 ( 4 x + 3 ) ^ { 2 } } = \frac { 1 } { 49 }$$

Solve for x

$$x=\frac{7\sqrt{857}-195}{64}\approx 0.155030256$$

$$x=\frac{-7\sqrt{857}-195}{64}\approx -6.248780256$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 98\left(4x+3\right)^{2}, the least common multiple of 2\left(4x+3\right)^{2},49.

$$49\left(1-3x\right)=2\left(4x+3\right)^{2}$$

Use the distributive property to multiply 49 by 1-3x.

$$49-147x=2\left(4x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+3\right)^{2}.

$$49-147x=2\left(16x^{2}+24x+9\right)$$

Use the distributive property to multiply 2 by 16x^{2}+24x+9.

$$49-147x=32x^{2}+48x+18$$

Subtract 32x^{2} from both sides.

$$49-147x-32x^{2}=48x+18$$

Subtract 48x from both sides.

$$49-147x-32x^{2}-48x=18$$

Combine -147x and -48x to get -195x.

$$49-195x-32x^{2}=18$$

Subtract 18 from both sides.

$$49-195x-32x^{2}-18=0$$

Subtract 18 from 49 to get 31.

$$31-195x-32x^{2}=0$$

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.

$$-32x^{2}-195x+31=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -32 for a, -195 for b, and 31 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-195\right)±\sqrt{\left(-195\right)^{2}-4\left(-32\right)\times 31}}{2\left(-32\right)}$$

Square -195.

$$x=\frac{-\left(-195\right)±\sqrt{38025-4\left(-32\right)\times 31}}{2\left(-32\right)}$$

Multiply -4 times -32.

$$x=\frac{-\left(-195\right)±\sqrt{38025+128\times 31}}{2\left(-32\right)}$$

Multiply 128 times 31.

$$x=\frac{-\left(-195\right)±\sqrt{38025+3968}}{2\left(-32\right)}$$

Add 38025 to 3968.

$$x=\frac{-\left(-195\right)±\sqrt{41993}}{2\left(-32\right)}$$

Take the square root of 41993.

$$x=\frac{-\left(-195\right)±7\sqrt{857}}{2\left(-32\right)}$$

The opposite of -195 is 195.

$$x=\frac{195±7\sqrt{857}}{2\left(-32\right)}$$

Multiply 2 times -32.

$$x=\frac{195±7\sqrt{857}}{-64}$$

Now solve the equation x=\frac{195±7\sqrt{857}}{-64} when ± is plus. Add 195 to 7\sqrt{857}.

$$x=\frac{7\sqrt{857}+195}{-64}$$

Divide 195+7\sqrt{857} by -64.

$$x=\frac{-7\sqrt{857}-195}{64}$$

Now solve the equation x=\frac{195±7\sqrt{857}}{-64} when ± is minus. Subtract 7\sqrt{857} from 195.

$$x=\frac{195-7\sqrt{857}}{-64}$$

Divide 195-7\sqrt{857} by -64.

$$x=\frac{7\sqrt{857}-195}{64}$$

The equation is now solved.

$$x=\frac{-7\sqrt{857}-195}{64} x=\frac{7\sqrt{857}-195}{64}$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 98\left(4x+3\right)^{2}, the least common multiple of 2\left(4x+3\right)^{2},49.

$$49\left(1-3x\right)=2\left(4x+3\right)^{2}$$

Use the distributive property to multiply 49 by 1-3x.

$$49-147x=2\left(4x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+3\right)^{2}.

$$49-147x=2\left(16x^{2}+24x+9\right)$$

Use the distributive property to multiply 2 by 16x^{2}+24x+9.

$$49-147x=32x^{2}+48x+18$$

Subtract 32x^{2} from both sides.

$$49-147x-32x^{2}=48x+18$$

Subtract 48x from both sides.

$$49-147x-32x^{2}-48x=18$$

Combine -147x and -48x to get -195x.

$$49-195x-32x^{2}=18$$

Subtract 49 from both sides.

$$-195x-32x^{2}=18-49$$

Subtract 49 from 18 to get -31.

$$-195x-32x^{2}=-31$$

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.

$$-32x^{2}-195x=-31$$

Divide both sides by -32.

$$\frac{-32x^{2}-195x}{-32}=-\frac{31}{-32}$$

Dividing by -32 undoes the multiplication by -32.

$$x^{2}+\left(-\frac{195}{-32}\right)x=-\frac{31}{-32}$$

Divide -195 by -32.

$$x^{2}+\frac{195}{32}x=-\frac{31}{-32}$$

Divide -31 by -32.

$$x^{2}+\frac{195}{32}x=\frac{31}{32}$$

Divide \frac{195}{32}, the coefficient of the x term, by 2 to get \frac{195}{64}. Then add the square of \frac{195}{64} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{195}{32}x+\left(\frac{195}{64}\right)^{2}=\frac{31}{32}+\left(\frac{195}{64}\right)^{2}$$

Square \frac{195}{64} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{195}{32}x+\frac{38025}{4096}=\frac{31}{32}+\frac{38025}{4096}$$

Add \frac{31}{32} to \frac{38025}{4096} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{195}{32}x+\frac{38025}{4096}=\frac{41993}{4096}$$

Factor x^{2}+\frac{195}{32}x+\frac{38025}{4096}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{195}{64}\right)^{2}=\frac{41993}{4096}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{195}{64}\right)^{2}}=\sqrt{\frac{41993}{4096}}$$

Simplify.

$$x+\frac{195}{64}=\frac{7\sqrt{857}}{64} x+\frac{195}{64}=-\frac{7\sqrt{857}}{64}$$

Subtract \frac{195}{64} from both sides of the equation.

$$x=\frac{7\sqrt{857}-195}{64} x=\frac{-7\sqrt{857}-195}{64}$$