Solve 4x^2-1/3-x=4

$$\frac { 4 x ^ { 2 } - 1 } { 3 - x } = 4$$

Solve for x

$$x = \frac{\sqrt{14} - 1}{2} \approx 1.370828693$$

$$x=\frac{-\sqrt{14}-1}{2}\approx -2.370828693$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by -x+3.

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

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

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

Add 4x to both sides.

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

Subtract 12 from both sides.

$$4x^{2}-1+4x-12=0$$

Subtract 12 from -1 to get -13.

$$4x^{2}-13+4x=0$$

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.

$$4x^{2}+4x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 4 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 4\left(-13\right)}}{2\times 4}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 4\left(-13\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-4±\sqrt{16-16\left(-13\right)}}{2\times 4}$$

Multiply -16 times -13.

$$x=\frac{-4±\sqrt{16+208}}{2\times 4}$$

Add 16 to 208.

$$x=\frac{-4±\sqrt{224}}{2\times 4}$$

Take the square root of 224.

$$x=\frac{-4±4\sqrt{14}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-4±4\sqrt{14}}{8}$$

Now solve the equation x=\frac{-4±4\sqrt{14}}{8} when ± is plus. Add -4 to 4\sqrt{14}.

$$x=\frac{4\sqrt{14}-4}{8}$$

Divide -4+4\sqrt{14} by 8.

$$x=\frac{\sqrt{14}-1}{2}$$

Now solve the equation x=\frac{-4±4\sqrt{14}}{8} when ± is minus. Subtract 4\sqrt{14} from -4.

$$x=\frac{-4\sqrt{14}-4}{8}$$

Divide -4-4\sqrt{14} by 8.

$$x=\frac{-\sqrt{14}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{14}-1}{2} x=\frac{-\sqrt{14}-1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by -x+3.

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

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

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

Add 4x to both sides.

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

Add 1 to both sides.

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

Add 12 and 1 to get 13.

$$4x^{2}+4x=13$$

Divide both sides by 4.

$$\frac{4x^{2}+4x}{4}=\frac{13}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{4}{4}x=\frac{13}{4}$$

Divide 4 by 4.

$$x^{2}+x=\frac{13}{4}$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{13}{4}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{13+1}{4}$$

Add \frac{13}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{7}{2}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{7}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{7}{2}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{14}}{2} x+\frac{1}{2}=-\frac{\sqrt{14}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{14}-1}{2} x=\frac{-\sqrt{14}-1}{2}$$