Solve 6x^2-29x-28quad1/2[x-1/2]=1/8

$$6 x ^ { 2 } - 29 x - 28 \quad \frac { 1 } { 2 } [ x - \frac { 1 } { 2 } ] = \frac { 1 } { 8 }$$

Solve for x

$$x = \frac{\sqrt{11869} + 115}{24} \approx 9.331039176$$

$$x=\frac{115-\sqrt{11869}}{24}\approx 0.252294157$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 8, the least common multiple of 2,8.

$$8\left(6x^{2}-29x\right)-4\left(28\times 2+1\right)\left(x-\frac{1}{2}\right)=1$$

Use the distributive property to multiply 8 by 6x^{2}-29x.

$$48x^{2}-232x-4\left(28\times 2+1\right)\left(x-\frac{1}{2}\right)=1$$

Multiply 28 and 2 to get 56.

$$48x^{2}-232x-4\left(56+1\right)\left(x-\frac{1}{2}\right)=1$$

Add 56 and 1 to get 57.

$$48x^{2}-232x-4\times 57\left(x-\frac{1}{2}\right)=1$$

Multiply 4 and 57 to get 228.

$$48x^{2}-232x-228\left(x-\frac{1}{2}\right)=1$$

Use the distributive property to multiply -228 by x-\frac{1}{2}.

$$48x^{2}-232x-228x+114=1$$

Combine -232x and -228x to get -460x.

$$48x^{2}-460x+114=1$$

Subtract 1 from both sides.

$$48x^{2}-460x+114-1=0$$

Subtract 1 from 114 to get 113.

$$48x^{2}-460x+113=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 48 for a, -460 for b, and 113 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-460\right)±\sqrt{\left(-460\right)^{2}-4\times 48\times 113}}{2\times 48}$$

Square -460.

$$x=\frac{-\left(-460\right)±\sqrt{211600-4\times 48\times 113}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-\left(-460\right)±\sqrt{211600-192\times 113}}{2\times 48}$$

Multiply -192 times 113.

$$x=\frac{-\left(-460\right)±\sqrt{211600-21696}}{2\times 48}$$

Add 211600 to -21696.

$$x=\frac{-\left(-460\right)±\sqrt{189904}}{2\times 48}$$

Take the square root of 189904.

$$x=\frac{-\left(-460\right)±4\sqrt{11869}}{2\times 48}$$

The opposite of -460 is 460.

$$x=\frac{460±4\sqrt{11869}}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{460±4\sqrt{11869}}{96}$$

Now solve the equation x=\frac{460±4\sqrt{11869}}{96} when ± is plus. Add 460 to 4\sqrt{11869}.

$$x=\frac{4\sqrt{11869}+460}{96}$$

Divide 460+4\sqrt{11869} by 96.

$$x=\frac{\sqrt{11869}+115}{24}$$

Now solve the equation x=\frac{460±4\sqrt{11869}}{96} when ± is minus. Subtract 4\sqrt{11869} from 460.

$$x=\frac{460-4\sqrt{11869}}{96}$$

Divide 460-4\sqrt{11869} by 96.

$$x=\frac{115-\sqrt{11869}}{24}$$

The equation is now solved.

$$x=\frac{\sqrt{11869}+115}{24} x=\frac{115-\sqrt{11869}}{24}$$

Steps for Completing the Square

Multiply both sides of the equation by 8, the least common multiple of 2,8.

$$8\left(6x^{2}-29x\right)-4\left(28\times 2+1\right)\left(x-\frac{1}{2}\right)=1$$

Use the distributive property to multiply 8 by 6x^{2}-29x.

$$48x^{2}-232x-4\left(28\times 2+1\right)\left(x-\frac{1}{2}\right)=1$$

Multiply 28 and 2 to get 56.

$$48x^{2}-232x-4\left(56+1\right)\left(x-\frac{1}{2}\right)=1$$

Add 56 and 1 to get 57.

$$48x^{2}-232x-4\times 57\left(x-\frac{1}{2}\right)=1$$

Multiply 4 and 57 to get 228.

$$48x^{2}-232x-228\left(x-\frac{1}{2}\right)=1$$

Use the distributive property to multiply -228 by x-\frac{1}{2}.

$$48x^{2}-232x-228x+114=1$$

Combine -232x and -228x to get -460x.

$$48x^{2}-460x+114=1$$

Subtract 114 from both sides.

$$48x^{2}-460x=1-114$$

Subtract 114 from 1 to get -113.

$$48x^{2}-460x=-113$$

Divide both sides by 48.

$$\frac{48x^{2}-460x}{48}=-\frac{113}{48}$$

Dividing by 48 undoes the multiplication by 48.

$$x^{2}+\left(-\frac{460}{48}\right)x=-\frac{113}{48}$$

Reduce the fraction \frac{-460}{48} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{115}{12}x=-\frac{113}{48}$$

Divide -\frac{115}{12}, the coefficient of the x term, by 2 to get -\frac{115}{24}. Then add the square of -\frac{115}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{115}{12}x+\left(-\frac{115}{24}\right)^{2}=-\frac{113}{48}+\left(-\frac{115}{24}\right)^{2}$$

Square -\frac{115}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{115}{12}x+\frac{13225}{576}=-\frac{113}{48}+\frac{13225}{576}$$

Add -\frac{113}{48} to \frac{13225}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{115}{12}x+\frac{13225}{576}=\frac{11869}{576}$$

Factor x^{2}-\frac{115}{12}x+\frac{13225}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{115}{24}\right)^{2}=\frac{11869}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{115}{24}\right)^{2}}=\sqrt{\frac{11869}{576}}$$

Simplify.

$$x-\frac{115}{24}=\frac{\sqrt{11869}}{24} x-\frac{115}{24}=-\frac{\sqrt{11869}}{24}$$

Add \frac{115}{24} to both sides of the equation.

$$x=\frac{\sqrt{11869}+115}{24} x=\frac{115-\sqrt{11869}}{24}$$