Solve (x+3)(6-x)/1-4x^2=-1/4

$$\frac { ( x + 3 ) ( 6 - x ) } { 1 - 4 x ^ { 2 } } = - \frac { 1 } { 4 }$$

Solve for x

$$x = \frac{\sqrt{155} + 3}{4} \approx 3.862474899$$

$$x=\frac{3-\sqrt{155}}{4}\approx -2.362474899$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{1}{2},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 4\left(2x-1\right)\left(2x+1\right), the least common multiple of 1-4x^{2},4.

$$-4\left(x+3\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply -4 by x+3.

$$\left(-4x-12\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply -4x-12 by 6-x and combine like terms.

$$-12x+4x^{2}-72=-\left(2x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply -1 by 2x-1.

$$-12x+4x^{2}-72=\left(-2x+1\right)\left(2x+1\right)$$

Use the distributive property to multiply -2x+1 by 2x+1 and combine like terms.

$$-12x+4x^{2}-72=-4x^{2}+1$$

Add 4x^{2} to both sides.

$$-12x+4x^{2}-72+4x^{2}=1$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$-12x+8x^{2}-72=1$$

Subtract 1 from both sides.

$$-12x+8x^{2}-72-1=0$$

Subtract 1 from -72 to get -73.

$$-12x+8x^{2}-73=0$$

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.

$$8x^{2}-12x-73=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -12 for b, and -73 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 8\left(-73\right)}}{2\times 8}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 8\left(-73\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-12\right)±\sqrt{144-32\left(-73\right)}}{2\times 8}$$

Multiply -32 times -73.

$$x=\frac{-\left(-12\right)±\sqrt{144+2336}}{2\times 8}$$

Add 144 to 2336.

$$x=\frac{-\left(-12\right)±\sqrt{2480}}{2\times 8}$$

Take the square root of 2480.

$$x=\frac{-\left(-12\right)±4\sqrt{155}}{2\times 8}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{155}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{12±4\sqrt{155}}{16}$$

Now solve the equation x=\frac{12±4\sqrt{155}}{16} when ± is plus. Add 12 to 4\sqrt{155}.

$$x=\frac{4\sqrt{155}+12}{16}$$

Divide 12+4\sqrt{155} by 16.

$$x=\frac{\sqrt{155}+3}{4}$$

Now solve the equation x=\frac{12±4\sqrt{155}}{16} when ± is minus. Subtract 4\sqrt{155} from 12.

$$x=\frac{12-4\sqrt{155}}{16}$$

Divide 12-4\sqrt{155} by 16.

$$x=\frac{3-\sqrt{155}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{155}+3}{4} x=\frac{3-\sqrt{155}}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{1}{2},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 4\left(2x-1\right)\left(2x+1\right), the least common multiple of 1-4x^{2},4.

$$-4\left(x+3\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply -4 by x+3.

$$\left(-4x-12\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply -4x-12 by 6-x and combine like terms.

$$-12x+4x^{2}-72=-\left(2x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply -1 by 2x-1.

$$-12x+4x^{2}-72=\left(-2x+1\right)\left(2x+1\right)$$

Use the distributive property to multiply -2x+1 by 2x+1 and combine like terms.

$$-12x+4x^{2}-72=-4x^{2}+1$$

Add 4x^{2} to both sides.

$$-12x+4x^{2}-72+4x^{2}=1$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$-12x+8x^{2}-72=1$$

Add 72 to both sides.

$$-12x+8x^{2}=1+72$$

Add 1 and 72 to get 73.

$$-12x+8x^{2}=73$$

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.

$$8x^{2}-12x=73$$

Divide both sides by 8.

$$\frac{8x^{2}-12x}{8}=\frac{73}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{12}{8}\right)x=\frac{73}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{3}{2}x=\frac{73}{8}$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{73}{8}+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{73}{8}+\frac{9}{16}$$

Add \frac{73}{8} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{155}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{155}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{155}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{155}}{4} x-\frac{3}{4}=-\frac{\sqrt{155}}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{\sqrt{155}+3}{4} x=\frac{3-\sqrt{155}}{4}$$