Solve sqrt{3a-4}=sqrt{2}a

$$\sqrt { 3 a - 4 } = \sqrt { 2 } a$$

Solve for a

$$a=\frac{3+\sqrt{23}i}{4}\approx 0.75+1.198957881i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{3a-4}\right)^{2}=\left(\sqrt{2}a\right)^{2}$$

Calculate \sqrt{3a-4} to the power of 2 and get 3a-4.

$$3a-4=\left(\sqrt{2}a\right)^{2}$$

Expand \left(\sqrt{2}a\right)^{2}.

$$3a-4=\left(\sqrt{2}\right)^{2}a^{2}$$

The square of \sqrt{2} is 2.

$$3a-4=2a^{2}$$

Subtract 2a^{2} from both sides.

$$3a-4-2a^{2}=0$$

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.

$$-2a^{2}+3a-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 3 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-3±\sqrt{3^{2}-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Square 3.

$$a=\frac{-3±\sqrt{9-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$a=\frac{-3±\sqrt{9+8\left(-4\right)}}{2\left(-2\right)}$$

Multiply 8 times -4.

$$a=\frac{-3±\sqrt{9-32}}{2\left(-2\right)}$$

Add 9 to -32.

$$a=\frac{-3±\sqrt{-23}}{2\left(-2\right)}$$

Take the square root of -23.

$$a=\frac{-3±\sqrt{23}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$a=\frac{-3±\sqrt{23}i}{-4}$$

Now solve the equation a=\frac{-3±\sqrt{23}i}{-4} when ± is plus. Add -3 to i\sqrt{23}.

$$a=\frac{-3+\sqrt{23}i}{-4}$$

Divide -3+i\sqrt{23} by -4.

$$a=\frac{-\sqrt{23}i+3}{4}$$

Now solve the equation a=\frac{-3±\sqrt{23}i}{-4} when ± is minus. Subtract i\sqrt{23} from -3.

$$a=\frac{-\sqrt{23}i-3}{-4}$$

Divide -3-i\sqrt{23} by -4.

$$a=\frac{3+\sqrt{23}i}{4}$$

The equation is now solved.

$$a=\frac{-\sqrt{23}i+3}{4} a=\frac{3+\sqrt{23}i}{4}$$

Substitute \frac{-\sqrt{23}i+3}{4} for a in the equation \sqrt{3a-4}=\sqrt{2}a.

$$\sqrt{3\times \frac{-\sqrt{23}i+3}{4}-4}=\sqrt{2}\times \frac{-\sqrt{23}i+3}{4}$$

Simplify. The value a=\frac{-\sqrt{23}i+3}{4} does not satisfy the equation.

$$-\frac{3}{4}\times 2^{\frac{1}{2}}+\frac{1}{4}i\times 46^{\frac{1}{2}}=\frac{1}{4}\times 2^{\frac{1}{2}}\left(-i\times 23^{\frac{1}{2}}+3\right)$$

Substitute \frac{3+\sqrt{23}i}{4} for a in the equation \sqrt{3a-4}=\sqrt{2}a.

$$\sqrt{3\times \frac{3+\sqrt{23}i}{4}-4}=\sqrt{2}\times \frac{3+\sqrt{23}i}{4}$$

Simplify. The value a=\frac{3+\sqrt{23}i}{4} satisfies the equation.

$$\frac{3}{4}\times 2^{\frac{1}{2}}+\frac{1}{4}i\times 46^{\frac{1}{2}}=\frac{1}{4}\times 2^{\frac{1}{2}}\left(3+i\times 23^{\frac{1}{2}}\right)$$

Equation \sqrt{3a-4}=\sqrt{2}a has a unique solution.

$$a=\frac{3+\sqrt{23}i}{4}$$