Solve 4sqrt{x-1}=3x-3

$$4 \sqrt{ x-1 } =3x-3$$

Solve for x

$$x = \frac{25}{9} = 2\frac{7}{9} \approx 2.777777778$$

$$x=1$$

Solution Steps

Square both sides of the equation.

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

Expand \left(4\sqrt{x-1}\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16\left(\sqrt{x-1}\right)^{2}=\left(3x-3\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$16\left(x-1\right)=\left(3x-3\right)^{2}$$

Use the distributive property to multiply 16 by x-1.

$$16x-16=\left(3x-3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-3\right)^{2}.

$$16x-16=9x^{2}-18x+9$$

Subtract 9x^{2} from both sides.

$$16x-16-9x^{2}=-18x+9$$

Add 18x to both sides.

$$16x-16-9x^{2}+18x=9$$

Combine 16x and 18x to get 34x.

$$34x-16-9x^{2}=9$$

Subtract 9 from both sides.

$$34x-16-9x^{2}-9=0$$

Subtract 9 from -16 to get -25.

$$34x-25-9x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-9x^{2}+34x-25=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-25. To find a and b, set up a system to be solved.

$$a+b=34 ab=-9\left(-25\right)=225$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 225.

$$1,225 3,75 5,45 9,25 15,15$$

Calculate the sum for each pair.

$$1+225=226 3+75=78 5+45=50 9+25=34 15+15=30$$

The solution is the pair that gives sum 34.

$$a=25 b=9$$

Rewrite -9x^{2}+34x-25 as \left(-9x^{2}+25x\right)+\left(9x-25\right).

$$\left(-9x^{2}+25x\right)+\left(9x-25\right)$$

Factor out -x in -9x^{2}+25x.

$$-x\left(9x-25\right)+9x-25$$

Factor out common term 9x-25 by using distributive property.

$$\left(9x-25\right)\left(-x+1\right)$$

To find equation solutions, solve 9x-25=0 and -x+1=0.

$$x=\frac{25}{9} x=1$$

Substitute \frac{25}{9} for x in the equation 4\sqrt{x-1}=3x-3.

$$4\sqrt{\frac{25}{9}-1}=3\times \frac{25}{9}-3$$

Simplify. The value x=\frac{25}{9} satisfies the equation.

$$\frac{16}{3}=\frac{16}{3}$$

Substitute 1 for x in the equation 4\sqrt{x-1}=3x-3.

$$4\sqrt{1-1}=3\times 1-3$$

Simplify. The value x=1 satisfies the equation.

$$0=0$$

List all solutions of 4\sqrt{x-1}=3x-3.

$$x=\frac{25}{9} x=1$$