Solve 4-x+24-6x/15x+39*114=14x

$$\frac { 4 - x + 24 - 6 x } { 15 x + 39 } \times 114 = 14 x$$

Solve for x

$$x = \frac{2 \sqrt{159} - 16}{5} \approx 1.843808085$$

$$x=\frac{-2\sqrt{159}-16}{5}\approx -8.243808085$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{13}{5} since division by zero is not defined. Multiply both sides of the equation by 3\left(5x+13\right).

$$\left(4-x+24-6x\right)\times 114=14x\times 3\left(5x+13\right)$$

Add 4 and 24 to get 28.

$$\left(28-x-6x\right)\times 114=14x\times 3\left(5x+13\right)$$

Combine -x and -6x to get -7x.

$$\left(28-7x\right)\times 114=14x\times 3\left(5x+13\right)$$

Use the distributive property to multiply 28-7x by 114.

$$3192-798x=14x\times 3\left(5x+13\right)$$

Multiply 14 and 3 to get 42.

$$3192-798x=42x\left(5x+13\right)$$

Use the distributive property to multiply 42x by 5x+13.

$$3192-798x=210x^{2}+546x$$

Subtract 210x^{2} from both sides.

$$3192-798x-210x^{2}=546x$$

Subtract 546x from both sides.

$$3192-798x-210x^{2}-546x=0$$

Combine -798x and -546x to get -1344x.

$$3192-1344x-210x^{2}=0$$

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.

$$-210x^{2}-1344x+3192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -210 for a, -1344 for b, and 3192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1344\right)±\sqrt{\left(-1344\right)^{2}-4\left(-210\right)\times 3192}}{2\left(-210\right)}$$

Square -1344.

$$x=\frac{-\left(-1344\right)±\sqrt{1806336-4\left(-210\right)\times 3192}}{2\left(-210\right)}$$

Multiply -4 times -210.

$$x=\frac{-\left(-1344\right)±\sqrt{1806336+840\times 3192}}{2\left(-210\right)}$$

Multiply 840 times 3192.

$$x=\frac{-\left(-1344\right)±\sqrt{1806336+2681280}}{2\left(-210\right)}$$

Add 1806336 to 2681280.

$$x=\frac{-\left(-1344\right)±\sqrt{4487616}}{2\left(-210\right)}$$

Take the square root of 4487616.

$$x=\frac{-\left(-1344\right)±168\sqrt{159}}{2\left(-210\right)}$$

The opposite of -1344 is 1344.

$$x=\frac{1344±168\sqrt{159}}{2\left(-210\right)}$$

Multiply 2 times -210.

$$x=\frac{1344±168\sqrt{159}}{-420}$$

Now solve the equation x=\frac{1344±168\sqrt{159}}{-420} when ± is plus. Add 1344 to 168\sqrt{159}.

$$x=\frac{168\sqrt{159}+1344}{-420}$$

Divide 1344+168\sqrt{159} by -420.

$$x=\frac{-2\sqrt{159}-16}{5}$$

Now solve the equation x=\frac{1344±168\sqrt{159}}{-420} when ± is minus. Subtract 168\sqrt{159} from 1344.

$$x=\frac{1344-168\sqrt{159}}{-420}$$

Divide 1344-168\sqrt{159} by -420.

$$x=\frac{2\sqrt{159}-16}{5}$$

The equation is now solved.

$$x=\frac{-2\sqrt{159}-16}{5} x=\frac{2\sqrt{159}-16}{5}$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{13}{5} since division by zero is not defined. Multiply both sides of the equation by 3\left(5x+13\right).

$$\left(4-x+24-6x\right)\times 114=14x\times 3\left(5x+13\right)$$

Add 4 and 24 to get 28.

$$\left(28-x-6x\right)\times 114=14x\times 3\left(5x+13\right)$$

Combine -x and -6x to get -7x.

$$\left(28-7x\right)\times 114=14x\times 3\left(5x+13\right)$$

Use the distributive property to multiply 28-7x by 114.

$$3192-798x=14x\times 3\left(5x+13\right)$$

Multiply 14 and 3 to get 42.

$$3192-798x=42x\left(5x+13\right)$$

Use the distributive property to multiply 42x by 5x+13.

$$3192-798x=210x^{2}+546x$$

Subtract 210x^{2} from both sides.

$$3192-798x-210x^{2}=546x$$

Subtract 546x from both sides.

$$3192-798x-210x^{2}-546x=0$$

Combine -798x and -546x to get -1344x.

$$3192-1344x-210x^{2}=0$$

Subtract 3192 from both sides. Anything subtracted from zero gives its negation.

$$-1344x-210x^{2}=-3192$$

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.

$$-210x^{2}-1344x=-3192$$

Divide both sides by -210.

$$\frac{-210x^{2}-1344x}{-210}=-\frac{3192}{-210}$$

Dividing by -210 undoes the multiplication by -210.

$$x^{2}+\left(-\frac{1344}{-210}\right)x=-\frac{3192}{-210}$$

Reduce the fraction \frac{-1344}{-210} to lowest terms by extracting and canceling out 42.

$$x^{2}+\frac{32}{5}x=-\frac{3192}{-210}$$

Reduce the fraction \frac{-3192}{-210} to lowest terms by extracting and canceling out 42.

$$x^{2}+\frac{32}{5}x=\frac{76}{5}$$

Divide \frac{32}{5}, the coefficient of the x term, by 2 to get \frac{16}{5}. Then add the square of \frac{16}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{32}{5}x+\left(\frac{16}{5}\right)^{2}=\frac{76}{5}+\left(\frac{16}{5}\right)^{2}$$

Square \frac{16}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{32}{5}x+\frac{256}{25}=\frac{76}{5}+\frac{256}{25}$$

Add \frac{76}{5} to \frac{256}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{32}{5}x+\frac{256}{25}=\frac{636}{25}$$

Factor x^{2}+\frac{32}{5}x+\frac{256}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{16}{5}\right)^{2}=\frac{636}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{16}{5}\right)^{2}}=\sqrt{\frac{636}{25}}$$

Simplify.

$$x+\frac{16}{5}=\frac{2\sqrt{159}}{5} x+\frac{16}{5}=-\frac{2\sqrt{159}}{5}$$

Subtract \frac{16}{5} from both sides of the equation.

$$x=\frac{2\sqrt{159}-16}{5} x=\frac{-2\sqrt{159}-16}{5}$$