Solve 5-sqrt{4x-3}=x

$$5 - \sqrt { 4 x - 3 } = x$$

Solve for x

$$x=7-\sqrt{21}\approx 2.417424305$$

Solution Steps

Subtract 5 from both sides of the equation.

$$-\sqrt{4x-3}=x-5$$

Square both sides of the equation.

$$\left(-\sqrt{4x-3}\right)^{2}=\left(x-5\right)^{2}$$

Expand \left(-\sqrt{4x-3}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{4x-3}\right)^{2}=\left(x-5\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{4x-3}\right)^{2}=\left(x-5\right)^{2}$$

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

$$1\left(4x-3\right)=\left(x-5\right)^{2}$$

Use the distributive property to multiply 1 by 4x-3.

$$4x-3=\left(x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$4x-3=x^{2}-10x+25$$

Subtract x^{2} from both sides.

$$4x-3-x^{2}=-10x+25$$

Add 10x to both sides.

$$4x-3-x^{2}+10x=25$$

Combine 4x and 10x to get 14x.

$$14x-3-x^{2}=25$$

Subtract 25 from both sides.

$$14x-3-x^{2}-25=0$$

Subtract 25 from -3 to get -28.

$$14x-28-x^{2}=0$$

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.

$$-x^{2}+14x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 14 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-14±\sqrt{196+4\left(-28\right)}}{2\left(-1\right)}$$

Multiply 4 times -28.

$$x=\frac{-14±\sqrt{196-112}}{2\left(-1\right)}$$

Add 196 to -112.

$$x=\frac{-14±\sqrt{84}}{2\left(-1\right)}$$

Take the square root of 84.

$$x=\frac{-14±2\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-14±2\sqrt{21}}{-2}$$

Now solve the equation x=\frac{-14±2\sqrt{21}}{-2} when ± is plus. Add -14 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}-14}{-2}$$

Divide -14+2\sqrt{21} by -2.

$$x=7-\sqrt{21}$$

Now solve the equation x=\frac{-14±2\sqrt{21}}{-2} when ± is minus. Subtract 2\sqrt{21} from -14.

$$x=\frac{-2\sqrt{21}-14}{-2}$$

Divide -14-2\sqrt{21} by -2.

$$x=\sqrt{21}+7$$

The equation is now solved.

$$x=7-\sqrt{21} x=\sqrt{21}+7$$

Substitute 7-\sqrt{21} for x in the equation 5-\sqrt{4x-3}=x.

$$5-\sqrt{4\left(7-\sqrt{21}\right)-3}=7-\sqrt{21}$$

Simplify. The value x=7-\sqrt{21} satisfies the equation.

$$7-21^{\frac{1}{2}}=7-21^{\frac{1}{2}}$$

Substitute \sqrt{21}+7 for x in the equation 5-\sqrt{4x-3}=x.

$$5-\sqrt{4\left(\sqrt{21}+7\right)-3}=\sqrt{21}+7$$

Simplify. The value x=\sqrt{21}+7 does not satisfy the equation because the left and the right hand side have opposite signs.

$$3-21^{\frac{1}{2}}=21^{\frac{1}{2}}+7$$

Equation -\sqrt{4x-3}=x-5 has a unique solution.

$$x=7-\sqrt{21}$$