Solve sqrt{-24x-23}-sqrt{32x-48}=1

$$\sqrt { - 24 x - 23 } - \sqrt { 32 x - 48 } = 1$$

Solve for x (complex solution)

$$x=\frac{-2\sqrt{26}i+22}{49}\approx 0.448979592-0.208123245i$$

Solution Steps

Subtract -\sqrt{32x-48} from both sides of the equation.

$$\sqrt{-24x-23}=1+\sqrt{32x-48}$$

Square both sides of the equation.

$$\left(\sqrt{-24x-23}\right)^{2}=\left(1+\sqrt{32x-48}\right)^{2}$$

Calculate \sqrt{-24x-23} to the power of 2 and get -24x-23.

$$-24x-23=\left(1+\sqrt{32x-48}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{32x-48}\right)^{2}.

$$-24x-23=1+2\sqrt{32x-48}+\left(\sqrt{32x-48}\right)^{2}$$

Calculate \sqrt{32x-48} to the power of 2 and get 32x-48.

$$-24x-23=1+2\sqrt{32x-48}+32x-48$$

Subtract 48 from 1 to get -47.

$$-24x-23=-47+2\sqrt{32x-48}+32x$$

Subtract -47+32x from both sides of the equation.

$$-24x-23-\left(-47+32x\right)=2\sqrt{32x-48}$$

To find the opposite of -47+32x, find the opposite of each term.

$$-24x-23+47-32x=2\sqrt{32x-48}$$

Add -23 and 47 to get 24.

$$-24x+24-32x=2\sqrt{32x-48}$$

Combine -24x and -32x to get -56x.

$$-56x+24=2\sqrt{32x-48}$$

Square both sides of the equation.

$$\left(-56x+24\right)^{2}=\left(2\sqrt{32x-48}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-56x+24\right)^{2}.

$$3136x^{2}-2688x+576=\left(2\sqrt{32x-48}\right)^{2}$$

Expand \left(2\sqrt{32x-48}\right)^{2}.

$$3136x^{2}-2688x+576=2^{2}\left(\sqrt{32x-48}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$3136x^{2}-2688x+576=4\left(\sqrt{32x-48}\right)^{2}$$

Calculate \sqrt{32x-48} to the power of 2 and get 32x-48.

$$3136x^{2}-2688x+576=4\left(32x-48\right)$$

Use the distributive property to multiply 4 by 32x-48.

$$3136x^{2}-2688x+576=128x-192$$

Subtract 128x from both sides.

$$3136x^{2}-2688x+576-128x=-192$$

Combine -2688x and -128x to get -2816x.

$$3136x^{2}-2816x+576=-192$$

Add 192 to both sides.

$$3136x^{2}-2816x+576+192=0$$

Add 576 and 192 to get 768.

$$3136x^{2}-2816x+768=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3136 for a, -2816 for b, and 768 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2816\right)±\sqrt{\left(-2816\right)^{2}-4\times 3136\times 768}}{2\times 3136}$$

Square -2816.

$$x=\frac{-\left(-2816\right)±\sqrt{7929856-4\times 3136\times 768}}{2\times 3136}$$

Multiply -4 times 3136.

$$x=\frac{-\left(-2816\right)±\sqrt{7929856-12544\times 768}}{2\times 3136}$$

Multiply -12544 times 768.

$$x=\frac{-\left(-2816\right)±\sqrt{7929856-9633792}}{2\times 3136}$$

Add 7929856 to -9633792.

$$x=\frac{-\left(-2816\right)±\sqrt{-1703936}}{2\times 3136}$$

Take the square root of -1703936.

$$x=\frac{-\left(-2816\right)±256\sqrt{26}i}{2\times 3136}$$

The opposite of -2816 is 2816.

$$x=\frac{2816±256\sqrt{26}i}{2\times 3136}$$

Multiply 2 times 3136.

$$x=\frac{2816±256\sqrt{26}i}{6272}$$

Now solve the equation x=\frac{2816±256\sqrt{26}i}{6272} when ± is plus. Add 2816 to 256i\sqrt{26}.

$$x=\frac{2816+256\sqrt{26}i}{6272}$$

Divide 2816+256i\sqrt{26} by 6272.

$$x=\frac{22+2\sqrt{26}i}{49}$$

Now solve the equation x=\frac{2816±256\sqrt{26}i}{6272} when ± is minus. Subtract 256i\sqrt{26} from 2816.

$$x=\frac{-256\sqrt{26}i+2816}{6272}$$

Divide 2816-256i\sqrt{26} by 6272.

$$x=\frac{-2\sqrt{26}i+22}{49}$$

The equation is now solved.

$$x=\frac{22+2\sqrt{26}i}{49} x=\frac{-2\sqrt{26}i+22}{49}$$

Substitute \frac{22+2\sqrt{26}i}{49} for x in the equation \sqrt{-24x-23}-\sqrt{32x-48}=1.

$$\sqrt{-24\times \frac{22+2\sqrt{26}i}{49}-23}-\sqrt{32\times \frac{22+2\sqrt{26}i}{49}-48}=1$$

Simplify. The value x=\frac{22+2\sqrt{26}i}{49} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Substitute \frac{-2\sqrt{26}i+22}{49} for x in the equation \sqrt{-24x-23}-\sqrt{32x-48}=1.

$$\sqrt{-24\times \frac{-2\sqrt{26}i+22}{49}-23}-\sqrt{32\times \frac{-2\sqrt{26}i+22}{49}-48}=1$$

Simplify. The value x=\frac{-2\sqrt{26}i+22}{49} satisfies the equation.

$$1=1$$

Equation \sqrt{-24x-23}=\sqrt{32x-48}+1 has a unique solution.

$$x=\frac{-2\sqrt{26}i+22}{49}$$