Solve sqrt{-4x+17}=x+1

$$\sqrt { - 4 x + 17 } = x + 1$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{-4x+17}\right)^{2}=\left(x+1\right)^{2}$$

Calculate \sqrt{-4x+17} to the power of 2 and get -4x+17.

$$-4x+17=\left(x+1\right)^{2}$$

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

$$-4x+17=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$-4x+17-x^{2}=2x+1$$

Subtract 2x from both sides.

$$-4x+17-x^{2}-2x=1$$

Combine -4x and -2x to get -6x.

$$-6x+17-x^{2}=1$$

Subtract 1 from both sides.

$$-6x+17-x^{2}-1=0$$

Subtract 1 from 17 to get 16.

$$-6x+16-x^{2}=0$$

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

$$-x^{2}-6x+16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-16=-16$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -6.

$$a=2 b=-8$$

Rewrite -x^{2}-6x+16 as \left(-x^{2}+2x\right)+\left(-8x+16\right).

$$\left(-x^{2}+2x\right)+\left(-8x+16\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(-x+2\right)+8\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(x+8\right)$$

To find equation solutions, solve -x+2=0 and x+8=0.

$$x=2 x=-8$$

Substitute 2 for x in the equation \sqrt{-4x+17}=x+1.

$$\sqrt{-4\times 2+17}=2+1$$

Simplify. The value x=2 satisfies the equation.

$$3=3$$

Substitute -8 for x in the equation \sqrt{-4x+17}=x+1.

$$\sqrt{-4\left(-8\right)+17}=-8+1$$

Simplify. The value x=-8 does not satisfy the equation because the left and the right hand side have opposite signs.

$$7=-7$$

Equation \sqrt{17-4x}=x+1 has a unique solution.

$$x=2$$